Soil air permeability and saturated hydraulic conductivity: Scale ...

SOIL AIR PERMEABILITY AND SATURATED 

HYDRAULIC CONDUCTIVITY: 

SCALE ISSUES, SPATIAL VARIABILITY, 

AND SURFACE RUNOFF MODELLING 

BO VANGSØ IVERSEN 

Environmental Engineering Laboratory 

Aalborg University 

Ph.D. Dissertation 2001

ISBN 87-90033-30-2 

ISSN 0909-6159: The Environmental Engineering Laboratory Ph.D. Dissertation Series

Soil air permeability and saturated hydraulic conductivity: Scale 

issues, spatial variability, and surface runoff modelling 

Ph.D. dissertation 

Luftpermeabilitet og mættet hydraulisk ledningsevne: Skalaforhold, 

rumlig variabilitet og modellering af overfladeafstrømning 

Ph.D.-afhandling 

Bo Vangsø Iversen 

Department of Environmental Engineering 

Institute of Life Sciences 

Aalborg University 

• 

Department of Crop Physiology and Soil Science 

Danish Institute of Agricultural Sciences 

1

Preface 

The present dissertation, which I have prepared at the Danish Institute of Agricultural Sci- 

ences (DIAS), is submitted in partial fulfilment of the requirement for the Doctor of Philoso- 

phy (Ph.D.) degree at Aalborg University (AAU). Associate Professor Per Moldrup, Depart- 

ment of Environmental Engineering, Institute of Life Sciences, AAU and Senior Scientist Per 

Schjønning, Department of Crop Physiology and Soil Science, DIAS have been my supervi- 

sors. 

The study was conducted during the period October 1997 to October 2001. The work 

was mainly carried out at the Department of Crop Physiology and Soil Science, DIAS. From 

October 1998 to April 1999 I also visited the Faculty of Geographical Sciences, Utrecht Uni- 

versity, the Netherlands, where I was working with Associate Professor Victor Jetten. 

I wish to thank my supervisors for their great help and inspiration during my study. 

Also I wish to thank Research Assistant Professor Tjalfe G. Poulsen and Consulting Engineer 

Per Loll for their co-operation. Victor Jetten taught me a lot about modelling and took good 

care of me during my stay in Utrecht. I would like to thank the technical staff at the Depart- 

ment of Crop Physiology and Soil Science (Michael Koppelgaard and Stig T. Rasmussen in 

particular) for their valuable work in the laboratory and in the field. I would also like to hon- 

our the memory of the late Senior Scientist Erik Sibbesen who sadly died in 1998. Erik was 

my local supervisor at the beginning of my Ph.D. project and he also offered me a job as a 

Scientist at DIAS back in 1996. Lastly, thank you to Astrid and Anita for being there for me. 

1 

Research Centre Foulum, October 2001 

Bo Vangsø Iversen

Summary 

Soil-physical properties such as water retention and air and water permeabilities (conductivi- 

ties) to a large extent govern the transport and fate of water, nutrients, and chemicals in soil 

systems. The quality and reliability of prediction by physical models describing subsurface 

and surface processes rely on the quality of the input data relating to the soil characteristics. 

The saturated hydraulic conductivity in particular is a sensitive parameter for hydraulic mod- 

els including surface runoff models. At the same time the variability of the saturated hydraulic 

conductivity is often high and the measuring methods are often time-consuming. Therefore 

accurate and faster methods of measuring the hydraulic characteristics of the soil are desirable 

to obtain a detailed knowledge of the spatial variability. The objectives of the present study 

were to develop and test a portable air permeameter, to investigate air and water permeability 

scale dependency, and to present predictive relationships between air permeability and satu- 

rated hydraulic conductivity. Finally, the objective was to investigate the spatial correlation 

structure of air permeability and to illustrate the use of air permeability as an input parameter 

in surface runoff modelling. 

The air permeability of a soil is defined as its ability to conduct air by the movement of 

molecules in response to a pressure gradient. The developed portable air permeameter was 

able to measure air permeability in situ, on-site and in the laboratory using two different sizes 

of core samples and gave reproducible results independent of sample size, when measuring in 

unstructured sandy soils. For structured loamy soils, sample size appeared to influence the 

results in accordance with the concept of a representative elementary volume. It was possible 

to carry out reliable in situ measurement of air permeability if a newly developed shape factor 

expression was applied. 

It appears not to be feasible to link air and water permeability using functional rela- 

tionships due to the different geometries and tortuosities of the gaseous and liquid phases. 

Other more empirical methods of linking air and water permeability might then be a better 

alternative. It seems likely that air permeability measured near field capacity will be a good 

prediction of the permeability of the entire pore system and thus a good prediction of the satu- 

rated hydraulic conductivity. The relation between air permeability (drained to a matric water 

potential of −50 or −100 cm H2O) and the saturated hydraulic conductivity measured on 100- 

cm 3 and 6280-cm 3 soil samples was examined and compared with an earlier presented rela- 

tionship. In general, there was a good relationship between log-transformed values for the 

studied soils. A poor relationship was found for a sandy soil having large number of medium- 

3

sized pores, illustrating the importance of adequate drainage of soil samples when measuring 

air permeability. Provided that the soil is adequately drained, the results indicated the exis- 

tence of a general log-log linear prediction relationship between air permeability and the satu- 

rated hydraulic conductivity independent of soil type and sample size. 

In the study, in situ measurements of air permeability were carried out in the topsoil in 

a 30-m grid in two small agricultural catchments using the developed portable air permeame- 

ter. The estimated semivariograms showed a spatial correlation of the log transformed values 

with a range of approximately 100 m. Additional on-site air permeability measurements 

(known boundary conditions) in an undisturbed constructed field in Japan indicated a spatial 

dependency of the log transformed data of approximately 20 m. The results from the studies 

therefore indicate that the portable air permeameter can be used to explain the spatial structure 

of the air permeability in an efficient and reliable way. This opens up for new methods of 

characterising the soil while obtaining new and valuable information about soil variability. 

A distributed model was used for simulations of surface runoff in two small agricul- 

tural catchments. A site-specific prediction relationship between air permeability and the satu- 

rated hydraulic conductivity was applied in the surface runoff model. The simulation results 

were highly dependent on whether the geometric average or kriged values of air permeability 

were used as model input. If only a few randomly chosen values of the saturated hydraulic 

conductivity were used to represent the spatial variation within the field slope, large devia- 

tions in repeated simulation results were obtained, both with respect to peak height and hy- 

drograph shape. On the other hand, when using many values of the saturated hydraulic con- 

ductivity based on an air permeability-saturated hydraulic conductivity relation, the model 

generally showed relatively comparable outputs, although simulations were sensitive to the 

chosen relation. Since massive measurement efforts will normally be required to get a satis- 

factory representation of spatial variability in saturated hydraulic conductivity, the use of in 

situ measurements of air permeability to assess this appears a promising alternative. 

4

Dansk resumé (Danish summary) 

Jordfysiske egenskaber såsom vandretention og luft- og vandpermeabilitet er i høj grad de 

styrende parametre i forbindelse med transporten af vand og næringsstoffer i jorden. Kvalite- 

ten og pålideligheden af fysiske modeller, der beskriver nær-overflade- og overfladeprocesser, 

er afhængige af kvaliteten af de data, modellen gør brug af. Den mættede hydrauliske led- 

ningsevne er en følsom parameter i de fleste hydrologiske modeller herunder også overflade- 

afstrømningsmodeller. Det er ydermere en parameter, der udviser en høj variabilitet samtidig 

med, at de anvendte målemetoder til brug for bestemmelsen er tidskrævende. Derfor er mere 

præcise og hurtigere målemetoder af jordens hydrauliske karakteristika ønskelige for at kunne 

opnå et bedre indblik i den rumlige variation i et givent område. Dette studie havde som for- 

mål at udvikle og teste et bærbart luftpermeameter, at undersøge skalaafhængigheden af luft- 

og vandpermeabilitet samt at præsentere prædiktive sammenhænge mellem luftpermeabilitet 

og mættet hydrauliske ledningsevne. Endeligt var formålet at undersøge luftpermeabilitetens 

rumlige korrelation og at illustrere dens anvendelighed som en input-parameter i relation til 

modellering af overfladeafstrømning. 

En jords luftpermeabilitet er defineret som dens evne til at lede luft ved bevægelse af 

molekyler under påvirkning af en trykgradient. Det fremstillede luftpermeameter, der var i 

stand til måle luftpermeabiliteten in situ, på stedet (on-site) og i laboratoriet på to forskellige 

prøvestørrelser, gav reproducerbare resultater, når der blev målt på sandede ustrukturerede 

jorde. For strukturerede mere lerede jorde påvirkede prøvevolumenet resultatet, hvilket var i 

overensstemmelse med konceptet for et repræsentativt prøvevolumen. Samtidig var det muligt 

at foretage pålidelige in situ-målinger af luftpermeabiliteten, hvor luftpermeabiliteten blev 

udregnet ved anvendelsen af et nyligt opstillet formfaktorudtryk. 

En funktionel sammenkædning af luft- og vandpermeabilitet virker tvivlsom pga. for- 

skellige geometrier og snoethed af gas- og væskefasen. Mere empiriske metoder til en sam- 

menkædning af de to faser kan derfor være et bedre alternativ. Det virker sandsynligt, at luft- 

permeabilitet målt ved at vandindhold nær feltkapacitet vil være et godt mål for permeabilite- 

ten i hele poresystemet og derved et godt bud på den mættede hydrauliske ledningsevne. Re- 

lationen mellem luftpermeabilitet (målt ved et vandpotentiale på −50 eller −100 cm H2O) og 

den mættede hydrauliske ledningsevne målt på henholdsvis 100 cm 3 og 6280 cm 3 ringprøver 

blev undersøgt og sammenlignet med en tidligere præsenteret sammenhæng. Generelt blev 

der på de undersøgte jorde fundet en god sammenhæng mellem log-transformerede værdier af 

luftpermeabilitet og mættet hydraulisk ledningsevne. En ringe sammenhæng blev fundet for 

5

en sandet jord med en høj frekvens af porer i mediumstørrelsen. Dette illustrerede vigtigheden 

af en tilstrækkelig afdræning af jordprøver i forbindelse med måling af luftpermeabilitet. På 

betingelse af, at jorden er tilstrækkelig afdrænet, viste resultaterne, at en log-log-lineær præ- 

diktiv sammenhæng mellem luftpermeabilitet og mættet hydrauliske ledningsevne sandsyn- 

ligvis eksisterer uafhængig af jordtype og prøvestørrelse. 

In situ-målinger af luftpermeabilitet udført med permeameteret blev udført i pløjelaget 

i et 30 gange 30 m net i to små oplande. Opstillede semivariogrammer viste, at der op til en 

afstand på omkring 100 m eksisterede en rumlig korrelation mellem de log-transformerede 

data. Supplerende målinger udført under kendte randbetingelser (on-site-målinger) på en kon- 

strueret jord i Japan viste samtidig, at der for denne jord op til en afstand på omkring 20 m 

eksisterede en rumlig korrelation mellem de log-transformerede data. Resultater for studierne 

var derfor med til at vise, at muligheden for at afdække luftpermeabilitetens rumlige variabili- 

tet på en hurtig og pålidelig måde eksisterer ved anvendelse af det udviklede udstyr. Dette 

åbner op for nye metoder til jordkarakterisering for på den måde at opnå ny og værdifuld in- 

formation omkring jordvariabilitet. 

En distribueret overfladeafstrømningsmodel blev brugt i forbindelse med simulerin- 

gerne. Til estimering af den mættede hydrauliske ledningsevne blev der anvendt en stedspeci- 

fik prædiktiv sammenhæng mellem luftpermeabilitet og mættet hydrauliske ledningsevne. 

Simuleringsresultaterne var i høj grad afhængig af, om det var den geometriske middelværdi 

eller de krigede værdier af luftpermeabiliteten, der blev brugt som input i modellen. Såfremt 

den rumlige variation udelukkende var repræsenteret af et lille antal tilfældigt udvalgte værdi- 

er af den mættede hydrauliske ledningsevne kunne store variationer i maksimal afstrømning 

og hydrografform indenfor gentagne simulationer konstateres. Omvendt viste resultaterne, at 

ved at anvende mange værdier af den mættede hydrauliske ledningsevne baseret på relationen 

mellem luftpermeabilitet og mættet hydraulisk ledningsevne, opnåede modellen relativt sam- 

menlignelige outputs, selvom simuleringerne var følsomme overfor den valgte relation. Da 

massive måleanstrengelser normalt vil være krævet for at opnå et tilfredsstillende billede af 

den rumlige variabilitet af den mættede hydrauliske ledningsevne, ser brugen af in situ- 

målinger af luftpermeabilitet ud til at være et lovende alternativ. 

6

List of supporting papers: 

I. B. V. Iversen, P. Schjønning, T. G. Poulsen, and P. Moldrup. In situ, on-site and labo- 

ratory measurements of soil air permeability: Boundary conditions and measurement 

scale. Soil Science 166 (2):97-106, 2001. 

II. B. V. Iversen, P. Moldrup, P. Schjønning, and P. Loll. Air and water permeability in 

differently-textured soils at two measurement scales. Soil Science 166(10):643-659, 

2001. 

III. B. V. Iversen, P. Moldrup, and P. Loll. Runoff modelling at two field slopes: use of in 

situ measurements of air permeability to characterise spatial variability of saturated 

hydraulic conductivity. Hydrological Processes (submitted), 2002. 

IV. T. G. Poulsen, B. V. Iversen, T. Yamaguchi, P. Moldrup, and P. Schjønning. Spatial 

and temporal dynamics of air permeability in a constructed field. Soil Science 166 

(3):153-162, 2001. 

7

Table of contents 

Preface........................................................................................................................................ 1 

Summary .................................................................................................................................... 3 

Dansk resumé (Danish summary) .............................................................................................. 5 

List of supporting papers............................................................................................................ 7 

Table of contents ........................................................................................................................ 9 

1 Introduction ........................................................................................................................... 11 

1.1 Heterogeneity and spatial variability in soil physical properties............................ 11 

1.2 Present barriers for near-surface hydrological modelling....................................... 14 

1.3 Objectives ............................................................................................................... 15 

2 Measuring air permeability ................................................................................................... 17 

2.1 Definition of air permeability ................................................................................. 17 

2.2 Development of a flexible, portable air permeameter ............................................ 17 

2.3 Test of air permeameter .......................................................................................... 20 

2.3.1 Test locations............................................................................................ 20 

2.3.2 Test results for repacked soils .................................................................. 20 

2.3.3 Test results for exhumed soil samples...................................................... 21 

2.3.4 Test results for shape factor (in situ use).................................................. 23 

2.4 Summary ................................................................................................................ 24 

3 Linking air and water permeability ....................................................................................... 25 

3.1 Air and water permeability ..................................................................................... 26 

3.2 Conceptually based correlation between air and water permeability ..................... 26 

3.3 Predictive relationships of small 100-cm 3 soil samples.......................................... 30 

3.4 Summary ................................................................................................................ 38 

4 Measurement scale ................................................................................................................ 39 

4.1 Representative elementary volume (REV) ............................................................. 39 

4.2 Air and water permeability at two measurement scales.......................................... 39 

4.2.1 Scaling behaviour of permeability............................................................ 44 

4.2.2 Prediction of saturated hydraulic conductivity at two scales ................... 46 

4.3 Summary ................................................................................................................ 47 

5 Spatial variability ................................................................................................................ 49 

5.1 Geostatistical analysis of air permeability .............................................................. 50 

5.2 Spatial variation of air permeability at two field slopes ......................................... 53 

9

5.3 Summary ................................................................................................................ 53 

6 Modelling surface runoff....................................................................................................... 55 

6.1 Key parameters and processes in relation to surface runoff ................................... 55 

6.2 LImburg Soil Erosion Model (LISEM) .................................................................. 57 

6.3 Modelling surface runoff at two field slopes .......................................................... 60 

6.3.1 Many measurements of air permeability versus few measure- 

ments of saturated hydraulic conductivity .............................................. 65 

6.4 Recommendation for using the proposed Ks-ka approach in runoff 

modelling............................................................................................................... 67 

6.5 Summary ................................................................................................................ 68 

7 Conclusions ........................................................................................................................... 69 

8 Perspectives........................................................................................................................... 71 

9 References ........................................................................................................................... 73 

10

1 Introduction 

Soil-physical properties such as water retention and air and water permeabilities (conductivi- 

ties) to a large extent govern the transport and fate of water, nutrients, and chemicals in soil 

systems. Most properties vary spatially as well as temporally. The spatial variability of the 

soil prevails in all three dimensions where it exhibits heterogeneity in topography, surface 

roughness, vegetation, and soil infiltration characteristics. This variability is a consequence of 

geology, geomorphological processes, and soil management. Since most physical parameters 

vary from place to place in almost every aspect, there are infinitely many places where a 

measurement would be desirable. In practice, it is only possible to sample a finite number 

which means that only a limited number of measurements of the soil physical variables (e.g. 

saturated hydraulic conductivity, texture, organic matter) is available. Consequently, spatial 

and temporal interpolation results in considerable uncertainties, which can have serious con- 

sequences when attempting to model hydrological processes in the soil system. 

1.1 Heterogeneity and spatial variability in soil physical properties 

Heterogeneity or spatial variability may be a composite of a deterministic and a stochastic 

component. The stochastic component in the nature is often assumed to be independent of 

position and can be described in several ways. The classical parametric estimators used to 

describe soil physical properties assume normality, randomness, and independence. Given a 

statistical population of a measured parameter, it is possible to estimate sample mean and 

sample variance. More completely, the statistical population can be defined by its probability 

density function, often characterised as a normal or a log-normal distribution. Flow-related 

properties such as the saturated hydraulic conductivity (Ks) have in several works been found 

to have a log-normal distribution (e.g. Nielsen et al., 1973; Baker and Bouma, 1976). Besides 

the randomness of the variability under certain circumstances it might as well be described as 

systematic. 

Intrinsic permeability is defined as the ability of a porous material to conduct a fluid in 

response to a pressure gradient and is known to vary systematically as well as randomly. Fig 

ure 1.1 illustrates how the intrinsic permeability of the soil may vary at different levels of 

scale. Besides the random variability, which cannot be related to a given cause, there may be 

several causes of variation at different scales. Within the soil profile (soil pedon) the intrinsic 

permeability may vary in depth, from the topsoil (e.g. a plough layer) to an illuvial low- 

permeable B horizon, caused by clay accumulation, cementation, or compaction of the hori- 

11

Figure 1.1. Possible causes of soil and air variability at different levels of scales. 

zon. Measurements of the permeability in the deepest horizon (C horizon) are believed to be a 

reflection of the properties of the parent material unaffected by the pedogenetic processes. 

Soil management may also be responsible for the variation of soil permeability. Heavy traffic 

in the agricultural fields leads to a compaction of the subsoil resulting in a less permeable soil 

layer. Within the same texture class, the intrinsic permeability may also vary because of dif- 

ferent levels of animal activity leading to a larger number of burrows in the soil. On a larger 

scale, the intrinsic permeability varies with soil type. I.e., whether it is an unstructured soil 

with a few macropores or a loamy structured or unstructured soil with or without preferential 

flow paths for the fluid. 

Structured variability can also be explained by an effect of sampling scale. Small vol- 

ume sampling often leads to a non-representative sampling of larger macropores resulting in a 

12

Figure 1.2. Measurement set-up and examination of spatial variability. (A) Principal outline 

of sampling scheme when measuring in a point using two different scales. (B) The complex of 

spatial variability, which can be expressed both as a structural and a random component 

(modified figure from Paper IV). (C) Outline and classification of the semivariogram (modi- 

fied figure from Paper III). 

lower estimate of permeability, compared to larger soil sample where a representative volume 

of soil is more likely to be sampled. 

13

The spatial variability of the intrinsic permeability of the soil may be examined at field 

scale when measuring in a transect or in a grid using various levels of measurement scale 

(Fig. 1.2A). The transect of the measured variable (Fig. 1.2B) often shows that the variable 

can be expressed as a structural and a random component and some spatially uncorrelated 

random noise. The statistical tool describing spatial variability is named geostatistics. Geosta- 

tistics assumes, unlike classical statistics, no sample independence. The basic tool of geosta- 

tistics is the semivariogram (Fig. 1.2C) which is used to interpret the spatial structure of the 

variable of interest. In geostatistics, observations taken at short distance are assumed to be 

more alike than observations made at points far apart (-that the values have a tendency to be 

correlated). Even though an element of randomness is still involved in the assumptions, this 

randomness is spatially correlated, known as the variation of the regionalised variable. The 

geostatistics tools are powerful in describing the spatial variability of a given parameter. 

1.2 Present barriers to near-surface hydrological modelling 

The quality and reliability of prediction by physical models describing subsurface and surface 

processes rely on the quality of the input data relating to the soil characteristics. With the rise 

in computing power and geographical information system (GIS) capabilities, spatially distrib- 

uted catchment models have been developed, which simulate the processes in larger and more 

complex catchments. The increased use of these models has increased the amount of data 

needed including knowledge of the spatial variability of soil physical parameters. 

Intensive and systematic sampling to obtain spatially distributed input data for large 

scale simulation models is often considered to be impossible because direct measurement of 

hydraulic properties is labour intensive, time-consuming, and thus expensive. Alternatively, 

these properties have to be estimated from available soil data. Estimation methods to obtain 

the required parameters from easily obtainable input data are called pedo-transfer functions 

(Bouma, 1989). Pedo-transfer functions serve to translate information found from easily 

measurable soil physical parameters into a form useful in broader applications, such as simu- 

lation modelling. The basic principle can be generalised to include any needed attribute that is 

not directly available, based on available data. A well-known example could be the estimation 

of the Ks by means of soil texture and/or content of organic matter (Schaap et al., 2001). 

The saturated hydraulic conductivity is a sensitive parameter for hydraulic models in- 

cluding surface runoff models. The variability of Ks is often large and the measuring methods 

are often time-consuming. Therefore accurate and faster methods of measuring the hydraulic 

characteristics of the soil are desirable to obtain a detailed knowledge of the spatial variabil- 

14

ity. Determination of Ks from more easily obtainable and/or readily available soil properties 

such as porosity or air permeability (ka) has been proposed in different studies (e.g. Giménez 

et al., 1997; Mckenzie and Jacquier, 1997; Poulsen et al., 1999; Loll et al., 1999; Timlin et 

al., 1999). 

Since Ks is a parameter dependent on both measuring method, scale of the measure- 

ment and spatial variability (Zobeck et al., 1985; Lauren et al., 1988; Rasmussen et al., 1993; 

Döll and Schneider, 1995; Messing and Jarvis, 1995; Mallants et al., 1996; Mallants et al., 

1997; Schulze-Makuch et al., 1999) several considerations have to be taken into account be- 

fore choosing the measurement method. Most often the choice of methods is not sufficient for 

reliable and representative Ks measurements. Therefore new concepts or methods have to be 

developed in order to incorporate representative infiltration parameters in distributed models. 

1.3 Objectives 

The objectives of the present study were: 

1. To develop and test a portable air permeameter capable of measuring air permeability (ka) 

in situ (with the sample still in place in the soil), on-site (exhumed soil samples), and in 

the laboratory using two different sizes of core samples (100 cm 3 and 3140 cm 3 or 6280- 

cm 3 , Papers I-IV). 

2. To investigate air and water permeability scale dependency measured in differently- 

textured soils using three different measurement scales (100 cm 3 , 3140 cm 3 , and 6280 

cm 3 , Papers I and II). 

3. To present predictive Ks (as a function of ka) relationships at two different measurement 

scales and to compare the results with earlier developed prediction relationships (Papers 

II and III). 

4. To investigate the spatial correlation structure of ka measured on undisturbed soils and to 

illustrate the use of ka as a water infiltration input parameter in relation to surface runoff 

modelling (Papers III and IV). 

15

2 Measuring air permeability 

2.1 Definition of air permeability 

Mass flow (also referred to as convective flow) is the movement of a fluid in response to a 

pressure gradient. The ability of a porous material (including soil) to conduct a fluid by this 

process is termed the intrinsic permeability (Reeve, 1953), which in theory is independent of 

the flowing fluid. The air permeability (ka) thus is an estimate of the intrinsic permeability 

when using gas as the flowing fluid. In this study, the intrinsic permeability will generally be 

referred to simply as the permeability. 

Air permeability has proven useful in the characterisation of soil pores (e.g. Groenevelt 

et al., 1984; Blackwell et al., 1990). Knowledge of ka and its variation with soil-water content 

is also necessary for modelling convective air and gas transport in soil, for example in relation 

to analysing and optimising soil vapour extraction systems for clean-up of soils contaminated 

with volatile organic compounds (Moldrup et al., 1998; Poulsen et al., 1999). 

At low pressure gradients, the flow of air through porous media is comparable to water 

flow. Fluid independent permeability (k) of the soil is most often estimated via Darcy’s law 

from measurement of air or water flow, according to 

⎛k ⎞⎛∆p⎞ q = ⎜ ⎟⎜ ⎟ 

⎝η⎠⎝∆x⎠ where q is flux density, η is the dynamic viscosity, p is pressure, and x is distance in flow 

direction. 

2.2 Development of a flexible, portable air permeameter 

Since in situ measurements are convenient and fast compared to laboratory measurements, it 

was decided to develop a portable air permeameter capable of measuring ka,in situ (i.e. with the 

sample still in place in the soil) and on-site (exhumed soil samples, ka,on-site, Paper I). The 

permeameter was constructed in order to allow measurements on both 100-cm 3 soil samples 

(only on exhumed soil samples) and on larger soil samples (3140 cm 3 and 6280 cm 3 , 20-cm 

inner diameter). 

The developed air permeameter is divided into six components as shown in Figure 2.1: 

1. Compressed air cylinder with pressure regulator for an approximate control of pres- 

sure. 

2. Two-stage regulator. 

17 

(1)

Figure 2.1. Apparatus for measuring soil air permeability in situ, on-site, and in the labora- 

tory using two ring sizes (100 cm 3 and 3140 cm 3 , figure from Paper I). 

3. A bank of three precision flow meters covering different flow ranges (0,2-2.3, 1.7- 

10.3, and 5.7-60 dm 3 /min). Each of these connected to a stopcock, allowing the air to 

flow exclusively in one of the flow meters. 

4. A water manometer 

5. A soil core adaptor for large soil cores (20-cm diameter) and a soil core adaptor for 

small soil cores (6.1-cm diameter). The former includes an inflatable rubber tube, 

which is inflated by a simple foot pump to seal the adaptor inside the sample ring. The 

small soil core adaptor is sealed to the sample ring by pressing a flexible rubber O-ring 

upward in the sample holder. The soil core adaptor for large soil cores can be used for 

measurements both in the field and in the laboratory, whereas the adaptor for small 

soil cores is designed for use in the laboratory. 

6. Hoses linking the flow meter bank, the adaptor and the manometer. 

18

The design of the instrument was partly based on earlier air permeameters used by 

Steinbrenner (1959), van Groenewoud (1968), Green and Fordham (1975), and Fish and 

Koppi (1994). 

Air permeability of exhumed soil samples (on-site or in the laboratory) is calculated 

using an integration of Eq. (1) (Kirkham, 1947), 

Q 

k ∆pa 

ηL 

a s = (2) 

s 

where Q is the volumetric flow rate, ∆p is the pressure difference across the sample, as is the 

cross-sectional area, and Ls is the length of the sample. 

When measuring ka,in situ, the air pressure at the lower end of the sample is not known 

because the air still has to flow through an (unknown) volume of soil before it reaches the soil 

surface (Fig. 2.1). The consequence of the lack of boundary conditions means that a “shape 

factor” has been introduced in the calculation of ka taking into account the geometry of the 

flow lines when the air leaves the lower part of the measuring cylinder in the soil (Grover, 

1955; Kirkham et al., 1958; Boedicker, 1972; Liang et al., 1995). As a result, Eq. (2) is reor- 

ganised by replacing as and Ls by the shape factor, A (Grover, 1955) 

Q 

k ∆pA 

η 

a = (3) 

The shape factor A may be regarded as an estimate of the as/Ls quotient in Eq. (2) in a meas- 

uring condition, where neither as or Ls involved in the flow is well defined. 

In the present study, A was determined using the finite element model (ANSYS F) 

developed by Liang et al. (1995) (Papers I and III). The shape factor equation of ANSYS F 

is 

A = 0.4862 

⎛D ⎞ 

0.0287 

⎛D ⎞ 

0.1106 

D ⎜ L ⎟− ⎜ − 

s L ⎟ 

⎝ ⎠ ⎝ s⎠ 

where D is the inside diameter of the soil core. 

2 

When introducing the shape factor into the calculation of ka the assumption of homo- 

geneity and isotropy in the soil is even more important because of the large, unknown volume 

of soil outside the measuring cylinder. Lower less permeable soil horizons and compacted 

layers (e.g. a plough pan) may violate the geometry of the flow lines leading to erroneous 

estimates of ka. Also highly structured soils with large amounts of macropores will affect the 

flow line geometry. 

19 

(4)

2.3 Test of air permeameter 

2.3.1 Test locations 

The newly developed air permeameter was tested and used in connection to the entire work in 

the present study (Papers I-IV). Figure 2.2 shows a map of Denmark showing the sites where 

measurements have been performed (Papers I-III). Besides the ten Danish locations, the air 

permeameter was used on a sandy loam in Higashi-Hiroshima, Japan (Paper IV, Fig. 2.2). 

Some general physical data of the soils are shown in Table 2.1. 

Table 2.1. Soils where newly developed air permeameter have been used. 

Site Soil type † Soil structure Horizons measured Paper 

Jyndevad Sand very weak/moderate Ap, Bhs I, II 

Silstrup Sandy clay loam coarse Ap, Bv I, II 

Lundgård Sand ‡ Ap I 

Foulum Loamy sand ‡ Ap I 

Fårdrup Sandy loam/Sandy clay 

loam 

weak/moderate Ap, Bvt I 

Slæggerup Sandy clay loam/Sandy coarse Ap, Bv I 

loam 

Estrup Sandy loam/Clay loam moderate/very coarse Ap, BE/Bhs/Bt, C II 

Tylstrup Sand weak Ap, Bv/Ap2, BC/C II 

Ans Sandy loam moderate/coarse Ap III 

Rødding Sandy loam moderate Ap III 

Higashi- 

Hiroshima 

Loam weak Ap IV 

† According to Soil Survey Division Staff (1993) 

‡ Sieved, packed soil was used in the experiment 

2.3.2 Test results for repacked soils 

The dependency of ka on sample size was tested in the laboratory on repacked soil samples 

using soil from the Foulum and Lundgård site (Paper I). Sieved and rewetted samples from 

each soil were packed at a pre-defined bulk density to a height of 10 cm in large steel cylin- 

ders with an inner diameter of 20 cm. The two soils were packed at three different water con- 

tents giving a total of six samples. Air permeability was measured on each soil sample using 

the newly developed air permeameter. Three small soil samples using 100-cm 3 steel cylinders 

with a diameter of 6.1 cm were subsequently taken inside each large ring and ka re-measured 

on these samples. The measurements of ka on the small and large repacked samples obtained 

in the experiment are compared in Figure 2.3. As expected for these homogenised samples, 

there was a good 1:1 relationship between the ka values obtained from the two sample types. 

20

Figure 2.2. Sites where measurements of ka in relation to the current work have been per- 

formed (Main map: Denmark, insert: Japan). 

It was then concluded that reliable measurements could be performed using the two sizes of 

soil samples when measuring in the laboratory. 

2.3.3 Test results for exhumed soil samples 

Measurements of ka were also carried out in situ using the portable permeameter (Paper I). 

To test the shape factor of Liang et al. (1995), field measurements were carried out at four 

different agricultural fields (Fårdrup, Jyndevad, Silstrup, and Slæggerup, Fig. 2.2). Measure- 

ments were carried out at three points in three different plots at each field. At each point, ka 

was measured in the Ap horizon (5-15 cm) and in the B horizon (approx. 35-45 cm) using the 

air permeameter. The large soil sample (20 cm diameter) was inserted to a depth of 10 cm in 

the soil and an in situ measurement of ka (ka,in situ) was performed. After that, the soil sample 

was exhumed from the soil, placed on a metal grid and ka,on-site was measured now with well- 

21

defined boundary conditions. Finally, three 100-cm 3 soil cores using the small rings were ex- 

tracted from each large ring. In the laboratory, the 100-cm 3 samples were weighed and ka was 

measured using the soil core adaptor for the small rings. The soil samples (100 cm 3 ) were 

then oven-dried at 105°C for 24 hours and weighed in order to determine soil bulk density and 

water content. 

Figure 2.3. Air permeability measured in the laboratory on repacked soils samples (100 cm 3 

and 3140 cm 3 ) on two soil types. Error bars show ± one standard error (n=3, figure from 

Paper I). 

k a (µm 2 ), small rings 

100 

50 

Lundgård 

Foulum 

1.1 

10 

10 50 

100 

k a (µm 2 ), large rings 

Figure 2.4 shows measurements of ka on two different soils (Silstrup and Jyndevad) 

using the two different sizes of soil samples. The two soils represent two extremes in a range 

of soil texture and structure (Table 2.1). From the figure it is obvious that there were large 

differences in ka between the two sample sizes from the structured loamy soil at Silstrup. 

However, there seemed to be a much better agreement between the two different sample sizes 

from the unstructured sandy soil at Jyndevad. The discrepancy between the two soil types is a 

reflection of a non-representative sampling of macropores in the structured soil using the 

small 100-cm 3 soil samples as discussed later (Section 4). However, the measurement in the 

unstructured sandy soils, which also resembled the test results in the laboratory for the re- 

packed soil samples, showed that there seemed to be a good agreement between the different 

sample sizes when measuring on undisturbed soil samples in an unstructured soil. 

22

k a [µm 2 ] 

100 

10 

1 

Large soil sample 

Small soil sample 

5% clay 

25% clay 

Ap B Ap B 

Jyndevad Silstrup 

Figure 2.4. Measurement of ka performed on two different soil types (Silstrup and Jyndevad) 

by using two different sizes of soil samples (100 cm 3 and 3140 cm 3 , data from Paper I). 

2.3.4 Test results for shape factor (in situ use) 

The relationship between ka measured with unknown (ka,in situ) and known boundary condi- 

tions (ka,on-site) is shown in Figures 2.5A and B. The two types of measurements compared 

well, especially for the A horizon. In situ measurement in the structured soils (Fårdrup, Sil- 

strup, and Slæggerup) generally compared less well 

than was the case for the less structured sandy soil at Jyndevad. 

The test of the finite element model of Liang et al. (1995) was based on comparing the 

agreement between results obtained with (on-site measurements) and without (in situ meas- 

urements using the shape factor) known boundary conditions at the lower end of the soil core 

(Paper I). Thus the test on the differences between in situ and on-site measurements consti- 

tutes a test on the applicability of the shape factor. Liang et al. (1995) developed and tested 

their shape factor on disturbed soils while we used undisturbed soils. A t-test indicated that 

there was no significant difference between the measurement methods in the A horizon, but in 

the B horizon a significant difference was detected (Paper I). However, it should be noted 

that the three extreme observations (Fig. 2.5B) belong to one site (Silstrup). The results there- 

fore indicated that the model of Liang et al. (1995) with caution might be applied also to 

structured soils in their undisturbed condition even though the assumptions of homogeneity 

and isotropy are ignored. 

23

k a,in situ (µm 2 ) 

k a,in situ (µm 2 ) 

1000 

100 

10 

1000 

100 

10 

a 

b 

Fårdrup 

Jyndevad 

Silstrup 

Slæggerup 

1:1 

Fårdrup 

Jyndevad 

Silstrup 

Slæggerup 

1:1 

1 

1 10 100 1000 

k a,on-site (µm 2 ) 

A horizon 

B horizon 

Figure 2.5. Air permeability measured in situ (ka,in situ) and on exhumed soil samples (ka,on-site) 

in the A and B horizon at four sites (figure from Paper I). 

2.4 Summary 

The air permeability of a soil is defined as its ability to conduct air by the movement of mole- 

cules in response to a pressure gradient. A portable air permeameter was constructed that was 

able to measure air permeability in situ, on-site, and in the laboratory using two different sizes 

of core samples. The newly developed device gave reproducible results independent of sam- 

ple size. For a structured loamy soil, sample size appeared to influence the results. It was pos- 

sible to carry out reliable in situ measurement of air permeability if the shape factor expres- 

sion developed by Liang et al. (1995) was applied. 

24

3 Linking air and water permeability 

As stated earlier, flow related properties such as Ks or ka are generally believed to have a log- 

normal distribution. In the present study, the measurement of Ks also indicated a log-normal 

distribution. Figure 3.1 shows histograms and the fitted log-normal distribution of Ks meas- 

ured on 100-cm 3 soil cores sampled in the plough layer at one of the studied soils (Rødding 

Field Slope). The symmetrics of the histograms and the general agreement with the fitted log- 

normal distribution seems to confirm that the measured Ks data follows a log-normal distribu- 

tion. However, a Shapiro-Wilk test also showed that it could be concluded with 95% confi- 

dence that the data distribution followed a log-normal distribution. Since it was believed that 

values of ka and Ks in this study followed a log-normal distribution, a log-transformation of 

the data was carried out before any statistical treatment. Since values of ka and Ks are also 

normally spread over several decades, the logarithmic scale is the best way to present the data. 

Frequency [%] 

14 

12 

10 

8 

6 

4 

2 

0 

Rødding 

0.1 1 10 100 

K s [m/d] 

Figure 3.1. Histograms and the fitted log-normal distribution of the Ks measured on 100 cm 3 

soil cores sampled in the plough layer at Rødding Field Slope. 

25 

log-normal distribution 

frequency

3.1 Air and water permeability 

The saturated hydraulic conductivity is an essential parameter in the analysis and modelling 

of water flow and chemical transport in the soil and expresses the capacity of a saturated soil 

to transmit water. Often, the parameter is used in predictive relationships for the unsaturated 

hydraulic conductivity based on the water retention characteristics (e.g. Campbell, 1974; 

Mualem, 1976; van Genuchten, 1980) where Ks is used as the reference point (Blackwell et 

al., 1990; Roseberg and McCoy, 1990). 

The relationship between the water permeability (kw) and Ks is described by 

g 

K k ρ 

w 

s = w 

(5) 

ηw 

where ρ is density, g is the gravitational acceleration, and subscripts of w indicate properties 

relating to water. 

Ideally, the permeability of air should be the same as that of water at similar fluid- 

phase contents (air or water). In reality, a perfect agreement between the two types of meas- 

urements would require that ka should be measured under totally dry conditions, but this 

would cause shrinkage of the soil leading to a breakdown of the soil structure. Inconsistency 

between the two types of measurements is also observed because air at atmospheric pressure 

does not act as a true fluid continuum in soils, so that the fluid velocity is not zero at solid 

boundaries as is the case with liquids (the Klinkenberg effect, Bear, 1972). Also, the two 

types of measurements differ because water as a polar fluid tends to interact with the amount 

of electrolyte in the water and the exchangeable cations in the soil causing a structural disrup- 

tion of the soil structure (Quirk, 1986). 

3.2 Conceptually based correlation between air and water permeability 

In spite of the earlier mentioned differences between ka and kw, Brooks and Corey (1964) 

developed functional relationships among saturation, pressure difference, and the permeabili- 

ties of air and water in terms of hydraulic properties of partially saturated porous media. 

These studies were based on measured data for repacked soils and showed relatively promis- 

ing results for simultaneous predictions of ka and kw. However, measurements on undisturbed 

26

k a /k a * 

k w /k w * 

1 

0.1 

0.01 

0.001 

0.0001 

10 

1 

0.1 

0.01 

0.001 

0.0001 

A 

0.06 0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1 

B 

ε a /ε a * 

0.06 0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1 

Volcanic Sand 

Fine Sand 

Fragmented Mixture 

Glass Beads 

Touchet Silt Loam 

ε w /ε w * 

Fragmented Fox Hill 

Berea Sandstone 

Hygiene Sandstone 

Lerbjerg 1 

Lerbjerg 3 

Lerbjerg 5 

Moldrup et al. 

(1998), η = 2 

M&Q (1961), η = 10/3 

Figure 3.2. (A) Relative air filled porosity (εa/εa*) as a function of relative air permeability 

(ka/ka*). (B) Relative water filled porosity (εw/εw*) as a function of relative water perme- 

ability (kw/kw*). Plotted are data from Brooks and Corey (1964) and Schjønning et al. (1999, 

air permeability data only). Also plotted, as straight lines are the models of Moldrup et al. 

(1998) and Millington and Quirk (1961). 

27

soils have shown that linking ka and kw is difficult due to different geometries and tortuosities 

of the gaseous and liquid phases (Moldrup et al., 2001). 

One way of describing and comparing the air and water transport parameters within 

different water or air contents is the Campbell (1974) constitutive model 

η 

p ⎛ α ⎞ 

= ⎜ ⎟ 

p* 

⎝α* ⎠ (6) 

where p is the transport parameter (e.g. ka or kw) and α is the fluid-phase content. p* is a cho- 

sen reference value of fluid-phase content α* , where α* in the present analysis is taken as the 

highest fluid-phase content where a parameter measurement is available. In a log(p/p*)- 

log(α/α*) co-ordinate system, Eq. (6) will yield a straight line with slope η, which is labelled 

a tortuosity factor describing the effect of tortuosity and other factors (e.g. connectivity, dead- 

end pores, and, in the case of water flow, water retention) on the change in the transport pa- 

rameter with fluid-phase content. 

Millington and Quirk (1961) and Moldrup et al. (1998) proposed values of η of 10/3 

and 2, respectively. Figures 3.2A and B show plots of p/p*(α/α*) for ka and kw measured on 

the repacked soils of Brooks and Corey (1964). In the work of Brooks and Corey (1964) ka 

and kw were measured in the same range of matric water potentials (saturation to a matric 

water potential of about −100 cm H2O) as in the present study. Also plotted in Figure 3.2A 

are soils (Lerbjerg 1, 3, and 5) sampled at three locations along a naturally occurring texture 

gradient with clay contents ranging from 11 to 46% (Schjønning et al., 1999). In addition, the 

proposed models of Millington and Quirk (1961) and Moldrup et al. (1998) are plotted as 

straight lines for both the ka and kw plots. Values of η for both fluids (air and water perme- 

abilities) and the ratio of η values are shown in Table 3.1. 

Measurements of ka on the repacked soil columns of Brooks and Corey (1964) showed 

values of η between 2 and 3, whereas the undisturbed Lerbjerg soils showed values of η be- 

tween 1 and 2. Highest values of η were found for the two sandstones. Air permeability rela- 

tions for the repacked soils were mostly placed within the models of Millington and Quirk 

(1961) and Moldrup et al. (1998), see Fig. 3.2A. The high value of η for the Berea Sandstone 

is probably related to an anisotropy of the medium. Generally, measurements of kw on the 

disturbed columns from Brooks and Corey (1964) showed significantly larger values of η 

compared to the ka measurements. Also, values were larger compared to the models of Mil- 

lington and Quirk (1961) and Moldrup et al. (1998). Values of η in relation to kw seemed to 

be more dependent on the soil type compared with ka. The relatively low values of η for the 

28

ka measurements are probably due to preferential air transport in the larger soil pores during 

convective air flow. 

Table 3.1. Tortuosity factor for convective air and water transport for selected soils. 

Site Tortuosity factor (air) Tortuosity factor (water) § 

Ratio of tortuosity 

(water/air) 

Volcanic Sand † 2.6 5.7 2.2 

Fine Sand † 2.4 4.4 1.8 

Fragmented Mixture † 2.7 6.5 2.4 

Glass Beads † 2.0 3.3 1.6 

Touchet Silt Loam † 2.1 6.1 2.9 

Fragmented Fox Hill † 2.7 -* - 

Berea Sandstone † 6.9 6.4 0.9 

Hygiene Sandstone † 3.0 11.8 4.0 

Lerbjerg 1 ‡ 1.2 no data - 



† Data from Brooks and Corey (1964) 

‡ Data from Schjønning et al. (1999) 

§ Water retention data interpolated from data connected to air permeability measurements of Brooks and Corey (1964) 

* Indication of double porosity 

This is not the case for kw where the effect of water retention to the soil particles cre- 

ates a much steeper decrease in kw with a decreasing ε. The best agreement between the two 

fluid-dependent tortuosity factors seems to be for the sandiest soils. Also it should be noted 

that for Fragmented Fox Hill η could not be estimated for water since data clearly showed 

dual-porosity behaviour for water but not for air (Fig. 3.2B). Figure 3.3 shows a plot of ηw 

and ηa. The figure shows a weak relation between the two parameters for the different porous 

media. This strongly implies that the general linking of ka and kw using reversed functional 

relationships as proposed by e.g. Brooks and Corey (1964) seems not to be feasible due to the 

different geometries and tortuosities of the gaseous and liquid phases. Using other and more 

empirical methods to link kw and ka at specific soil-water matric potentials may therefore be a 

better alternative. 

29

η a 

8 

7 

6 

5 

4 

3 

2 

1 

0 10 20 30 40 

η w 

Volcanic Sand 

Fine Sand 

Fragmented Mixture 

Glass Beads 

Touchet Silt Loam 

Berea Sandstone 

Hygiene Sandstone 

Fragmented Fox Hill 

Figure 3.3. Plot of ηw in relation to ηa using data from Brooks and Corey (1964). 

3.3 Predictive Ks-ka relationships for small 100-cm 3 soil samples 

The physical relationship between ka and Ks indicated by Eq. (2) and (5) renders it probable 

that an empirical Ks-ka relationship exists. The flow rate of a fluid in the fluid-filled, continu- 

ous pores depends on the fourth power of the effective pore radius according to Pouseuille´s 

law (Hillel, 1998). When the soil is drained to or near field capacity (e.g. to a matric water 

potential of −50 to −100 cm H2O) the flow of air will take place in the large spectrum of soil 

pores (>30-60 µm). Therefore it seems likely that ka measured near field capacity will be a 

good prediction of the permeability of the entire pore system and thus a good prediction of 

Ks. However, only few studies have presented or used this kind of measurement to present 

dynamically based prediction relationships of Ks from measurements of ka (Schjønning, 

1986; Riley and Ekeberg, 1989; Blackwell et al., 1990; Rasmussen et al., 1993; Riley and 

Eltun, 1994; Loll et al., 1999). 

Since measurements of Ks are time-demanding and the quality of the measurements 

often not proportional with the amount of time used, ka measurements could be used to de- 

termine values of Ks. Even though the Ks(ka) prediction relationship can have an accuracy 

more than plus/minus one order of magnitude, it may be preferable to use ka as it is rapid and 

non-destructive, and poses fewer practical problems compared to measurement of Ks 

(Kirkham, 1947; Grover, 1955; Janse and Bolt, 1960; Blackwell et al., 1990; Fish and Koppi, 

1994). Measurements of Ks are often impossible to accomplish at more than a few sites with a 

30

limited budget and therefore a detailed knowledge of the spatial variability becomes impossi- 

ble. Using measurement of ka then increases significantly the number of measurements within 

the same budget 

log K s [m/d] 

2 

1 

0 

-1 

-2 

-3 

y = 0.72x + 8.78 

r 2 = 0.92 

log-log linear prediction (Loll et al., 1999) 

95% pred. interv. (Loll et al., 1999) 

log-log linear prediction 

95% prediction interval 

-13 -12 -11 -10 

Volcanic Sand (-137 cm H 2 O) 

Fine Sand (-92 cm H 2 O) 

Fragmented Mixture (-54 cm H 2 O) 

Glass Beads (-150 cm H 2 O) 

Touchet Silt Loam (-98 cm H 2 O) 

log k a [m 2 ] 

Fragmented Fox Hill (-58 cm H 2 O) 

Berea Sandstone (-85 cm H 2 O) 

Hygiene Sandstone (-101 cm H 2 O) 

Figure 3.4. Relation between Ks and ka measured on repacked soil samples. Data from 

Brooks and Corey (1964). The 95% prediction interval, n, α, β, and r 2 are given as well. 

Numbers in parentheses (cm H2O) are the matric water potentials at which ka was measured. 

Also plotted is the relationship and 95% prediction interval between ka (measured at a matric 

water potential of −100 cm H2O) and Ks found by Loll et al. (1999) on 100-cm 3 soil samples. 

In the work of Papers II and III it was assumed that a correlation exists between 

log(Ks) and log(ka), i.e. 

log ( K ) = α log ( k ) + β 

(7) 

s a 

Loll et al. (1999) explored the existence of a general prediction relationship between ka 

(measured at a matric water potential at −100 cm H2O) and Ks. The data used were measure- 

ments of ka and Ks on 100-cm 3 soil samples taken from nine different data sets representing a 

variety of soil types (Schjønning, 1986; Riley and Ekeberg, 1989). The general log-log linear 

31

prediction relationship combining the nine data sets (n=1614) had a prediction accuracy of 

±0.7 orders of magnitude. 

i.e. α=1.27 and β=14.11 in Eq. (7). 

2 

log( Ks)[ m/ d] 1.27log( ka)[ m ] 14.11 

= + (8) 

In Figure 3.4, data of ka and kw from Brooks and Corey (1964) are shown in a log(ka)- 

log(Ks) plot. In order to compare data with the work of Loll et al. (1999), values of ka drained 

to a matric water potential near −100 cm H2O have been chosen. The general prediction rela- 

tionship of Loll et al. (1999) are shown in the figure as well. Even though values of ka predict 

values of Ks well (r 2 =0.92), there seems to be a poor agreement with the prediction relation- 

ship of Loll et al. (1999). The poor agreement between the two predictions is most likely re- 

lated to the fact that Brooks and Corey (1964) measured on repacked soils whereas Loll et al. 

(1999) measured on undisturbed soil samples. This illustrates the importance of distinguishing 

between these two extremities of measurement conditions (repacked or undisturbed), but also 

that a reasonable prediction relationship between ka and kw might exist for both of the two 

measuring conditions. Figure 3.4 also illustrates that since the traditional linking of ka and kw 

presented by e.g. Brooks and Corey (1964) is questioned, it would be more logical to find a 

reference point where larger pores control both fluid flows. A linking between ka (drained to 

a matric water potential at −50 or −100 cm H2O) and Ks might then be a better alternative. It 

was therefore decided to further examine the relation between the two parameters in the pre- 

sent study. 

In the work of Paper II, measurements of ka (drained to matric water potential at −50 

cm H2O) and Ks on 100-cm 3 soil cores were carried out on four different Danish agricultural 

soils (Table 2.1, Fig. 2.2) ranging from sand to loam (Soil Survey Division Staff, 1993). Site 

specific relationships for the soil cores are shown in Figure 3.5. Also plotted in the figure is 

the relationship of Loll et al. (1999). The prediction relationship between the loamy soils (Sil- 

strup and Estrup) seemed to display visual similarities. The 95% prediction interval for both 

sites indicated accuracy better than ±1.7 orders of magnitude. Compared to Silstrup and Es- 

trup, the values of α (slope) appeared lower for the sandy soil at Jyndevad. However, the 95% 

prediction interval indicated an accuracy of around ±0.4 orders of magnitude. A poor predic- 

tion relationship was revealed for the other sandy soil (Tylstrup), where a distinct difference 

between the uppermost horizon (Ap) and the two deepest horizons (B and BC/C) was ob- 

served. The large differences in the prediction relationships between the two sandy soils are 

32

log(K s ) [m/d] 


4 

3 

n = 49 

y = 1.35x + 15.20 

2 

1 

0 

-1 

-2 

r 

-3 

-4 

Silstrup Estrup 

4 

3 

2 

1 

0 

-1 

-2 

-3 

-4 

Jyndevad 

Tylstrup 

-14 -13 -12 -11 -10-14 

-13 -12 -11 -10 

2 n = 41 

y = 1.27x + 14.55 

= 0.63 

r 2 = 0.62 

n = 81 

y = 1.10x + 12.57 

r 2 n = 53 

y = 0.48x + 5.85 

= 0.81 

r 2 A 

B 

C D 

= 0.05 

log(k a ) [m 2 ] 

Ap, Profile 1 

Ap, Profile 2 

Ap, Profile 3 

B, Profile 1 

B, Profile 2 

B, Profile 3 

log(k a ) [m 2 ] 




95% prediction interval (Loll et al., 1999) 

BC/C, Profile 1 



Figure 3.5. Log-log linear prediction relationship between ka (measured at a matric water 

potential of −50 cm H2O) and Ks measured on the individual 100-cm 3 samples for each of the 

four sites. The 95% prediction interval, n, α, β, and r 2 are given. Also plotted is the relation- 

ship and 95% prediction interval between ka (measured at a matric water potential of −100 

cm H2O) and Ks found by Loll et al. (1999) on 100-cm 3 soil samples (modified figure from 

Paper II). 

probably explained by the differences in the soil water characteristics. This is exemplified in 

Figure 3.6, which shows the pore size distributions of the studied soils derived from water 

retention data. The figure describes the frequency of pores of different size on a logarithmic 

scale. The calculation of the frequency curves are based on the fact that the matric water po- 

tential is related to the effective pore diameter and that the first derivative of the soil water 

33

Pore volume per 1/10 

pF value [% vol/vol] 







4 

3 

2 

1 

0 

4 

3 

2 

1 

0 

4 

3 

2 

1 

0 

4 

3 

2 

1 

Silstrup Ap 

Estrup Ap 

Jyndevad Ap 

Tylstrup Ap1 

A Silstrup Bv B Silstrup 

BC(g)/Cc 

C 

D Estrup 

E Estrup 

F 

BE(g)/Bt(g)/ 

Bhs 

Cg/Cc/C 

G Jyndevad H Jyndevad I 

Bhs/Bs 

BC/C 

J Tylstrup K Tylstrup 

L 

Bv/Ap2 

BC/C 

0 

5 4 3 2 1 0 5 4 3 2 1 0 5 4 3 2 1 0 pF 

0.03 0.3 3 30 300 3000 0.03 0.3 3 30 300 3000 0.03 0.3 3 30 300 3000 D 

Figure 3.6. Pore size distribution (Ap, B, and C horizon) calculated from water retention data 

(arithmetic average) assuming D=3000/-ψ (D=tube equivalent pore diameter, µm) where ψ 

is the matric water potential in cm H2O. Ordinate is percentage of pore volume per 1/10 log(- 

ψ) values (m 3 100m -3 ). The dashed vertical lines mark −50 cm H2O (pF 1.7), the matric water 

potential at which ka was measured. It should be noted that a swelling tendency was observed 

for the soil in the B and C horizons at Estrup. Therefore the distributions were calculated 

using a constructed value of the bulk density in order to obtain a meaningful soil water char- 

acteristic curve (figure from Paper II). 

34

characteristic expresses the frequency of pores (Schjønning, 1992). The distinct peaks of pore 

volume for the two sandy soils (Jyndevad and Tylstrup) are a result of a well-sorted particle 

size distribution typical for glaciofluvial (Jyndevad) and postglacial marine (Tylstrup) sedi- 

ments. The striking difference between the two soils is that the distinct peak of pore volume 

for the Jyndevad soil is on the dry side of the matric water potential to which the soil samples 

were drained (pF 1.7) when measuring ka. The peak for the Tylstrup soil, on the other hand, 

is on the wet side. In other words, while the soil at Jyndevad had drained most of its pores at 

pF 1.7, the soil at Tylstrup still contained a large number of water-filled pores at this matric 

water potential. The relatively high variability for the ka measurements (compared to Ks) in 

the two deepest horizons for the Tylstrup soil was then probably a result of the steep part of 

the pore size distribution curve being exactly at pF 1.7 (Figures 3.6K and L). Even a small 

deviation in pF values around 1.7 resulted in large changes in the water contents between the 

individual sample and a corresponding high variability in ka measurements. Unlike ka meas- 

urements, measurements of Ks reflected the entire pore continuum having a correspondingly 

low variability. For the uppermost horizon, the picture was less clear. The pore size distribu- 

tion curve was less steep at pF 1.7 compared to the two other horizons. Here a high variability 

for Ks measurements was reflected in a correspondingly high variability for measurements of 

ka. 

In order to evaluate the performance of the regressions of the prediction relationships 

and to compare them with the general prediction relationship of Loll et al. (1999), the 95% 

confidence interval was calculated using a bias-corrected and accelerated (BCa) percentile 

bootstrapping method (Efron and Tibshirani, 1993). The calculations were only done for the 

Silstrup, Estrup, and Jyndevad sites. The Tylstrup site was left out because of the insufficient 

drainage of the soil cores. The BCa percentile bootstrapping and a visual examination of Fig- 

ure 3.5 confirmed that, with the small 100-cm 3 -soil samples, only the relationship of the Jyn- 

devad site had a significantly different slope (α) compared to the general relationship of Loll 

et al. (1999). Also the BCa percentile bootstrapping estimates of the average log(Ks) pre- 

dicted from the general relationship of Loll et al. (1999) did not show any significant differ- 

ences when compared to the average log(Ks) predicted from the site-specific relationships. 

In Figure 3.7A the prediction relationship between Ks and ka is plotted. Samples are 

drained to a matric water potential of −50 cm H2O and bulked together for three of the soils 

from Paper II (Silstrup, Estrup, and Jyndevad). Because of the insufficient drainage, the 

sandy soil at Tylstrup was left out. In Paper III additional measurements of Ks and ka were 

also carried out on 100-cm 3 soil cores sampled on a sandy loam in a 30-m grid in a small 

35



3 

2 

1 

0 

-1 

-2 

-3 

-4 

3 

2 

1 

0 

-1 

-2 

y = 1.29x + 14.55 

r 2 = 0.77 

y = 1.38x + 15.13 

r 2 = 0.54 

-3 

Ans Field Slope 

-4 

-14 -13 -12 -11 -10 

log(k a ) [m 2 ] 

Silstrup 

Estrup 

Jyndevad 





A 

B 

0 cm 

24 cm 

50 cm 

Figure 3.7. (A) Log-log linear prediction relationship between ka measured at a matric water 

potential of −50 cm H2O and Ks measured on 100-cm 3 soil samples from three sites. The 95% 

prediction interval, n, α, β, and r 2 are given (modified figure from Paper II). (B) Log-log lin- 

ear prediction relationship between ka (measured at a matric water potential of −100 cm 

H2O) and Ks measured on 100-cm 3 soil samples from Ans Field Slope at three depths. 

36

agricultural catchment (2.8 hectares) named Ans Field Slope (Figure 2.2). In this case, ka was 

measured on samples drained to a matric water potential of −100 cm H2O. Soil samples were 

taken from three different depths: 0 cm, approximately 25 cm, and 50 cm. The site-specific 

log-log linear prediction relationship for the soils at Ans Field Slope is presented in Figure 

3.7B. In both figures, the relationship established by Loll et al. (1999) is shown as well. 

The 95% prediction interval related to the site-specific log-log linear prediction rela- 

tionship on the 100-cm 3 soil samples in Paper II indicated a relatively low level of accuracy, 

close to ±1.2 orders of magnitude (Fig. 3.7A). 

2 

log( Ks)[ m/ d] = 1.29log( ka)[ m ] + 14.55 

The best-fit prediction relationship, Eq. (9), matched the relationship by Loll et al. (1999) 

well. The statistical analyses using the BCa percentile bootstrapping method revealed that α 

for the small samples had a value corresponding closely to the value of α for the general rela- 

tionship of Loll et al. (1999). The average value of log(Ks) predicted from the general predic- 

tion relationship also compared well with the relationship of Loll et al. (1999). 

Even though a large scatter was seen for the studied soils in Paper III (accuracy 

around ±1.8 orders of magnitude, Fig. 3.7B) the prediction relationship, Eq. (10) 

2 

log( K )[ m/ d] = 1.38log( k )[ m ] + 15.13 

(10) 

s a 

fitted well with the general relationship of Loll et al. (1999). The 95% confidence interval 

calculated using the BCa percentile bootstrapping method confirmed that no significant dif- 

ference in the linear prediction relationship existed between the individual depths, between 

the individual depths and the three depths bulked together, and between the three depths 

bulked together and the general relationship of Loll et al. (1999). The prediction relationship 

shown in Figure 3.7B omits the two most extreme observations (marked with a circle), which 

gave an accuracy of the prediction relationship of ±1.2 orders of magnitude, the same level as 

the accuracy of the prediction relationship in Paper II (Fig. 3.7A). 

If the soil is sufficiently drained, the work of Papers II and III indicates that a general 

log-log linear prediction relationship between ka and Ks appears to exist independent of soil 

type. Although the soil samples used by Loll et al. (1999) were drained to a matric water po- 

tential of −100 cm H2O, the difference between the two relationships seemed to be minimal. 

This confirms that it is the largest pores of the soil that almost exclusively are active in the 

transport of air in accordance with Pouseuille´s law. The opening up of pores in the interval of 

60 to 30 µm when draining the soil from a matric water potential of −100 cm H2O to −50 cm 

37 

(9)

H2O had only little effect on the value of ka. The number of replicates seemed to be much 

more important in the determination of a general prediction relationship. 

Earlier studies of Ks determinations in undisturbed soils from more easily obtainable 

soil properties (Ahuja et al., 1984; Giménez et al., 1997; Poulsen et al., 1999) revealed that 

the prediction accuracy for Ks based on static soil characteristics generally was ±1 order of 

magnitude or worse. Even though the prediction accuracy between ka and Ks found in this 

study in general was higher than ±1 order of magnitude, this should be compared to the per- 

formance of the in situ measurements of ka. During the same period, many measurements of 

ka can be carried out compared to only a few of Ks. Ks is an extremely variable parameter 

showing variation sometimes in excess of three orders of magnitude (Mohanty et al., 1994). 

Therefore this study opens up for an alternative way of exploring the spatial variability of the 

infiltration parameter in an area through measurements of ka. 

3.4 Summary 

The linking of ka and kw using functional relationships seems not to be feasible due to the 

different geometries and tortuosities of the gaseous and liquid phases. Using other more em- 

pirical methods of linking kw and ka might then be a better alternative. 

The flow rate of a fluid in the fluid-filled, continuous pores depends on the fourth 

power of the effective pore radius according to Pouseuille´s law. When the soil is drained to 

or near field capacity, the flow of air will take place in the large soil pores (>30-60 µm). 

Therefore it seems likely that ka measured near field capacity will be a good prediction of the 

permeability of the entire pore system and hence a good prediction of Ks. In the work of Pa- 

per II and Paper III, the relation between ka (drained to a matric water potential of −50 or 

−100 cm H2O) and Ks measured on 100-cm 3 soil samples was examined and compared with 

an earlier relationship presented by Loll et al. (1999). In general, a good relationship between 

log(Ks) and log(ka) was found for the studied soils, which compared well with the relation- 

ship of Loll et al. (1999). A poor relationship was found for a sandy soil having a high fre- 

quency of medium-sized pores, illustrating the importance of the drainage of the soil samples 

when measuring ka. 

If the soil is sufficiently drained, the results indicate that a general log-log linear pre- 

diction relationship between ka and Ks exists independent of soil type. This opens up for an 

alternative way of exploring the spatial variability of the infiltration parameter in an area 

through measurements of ka instead of Ks. 

38

4 Measurement scale 

4.1 Representative elementary volume (REV) 

A point measurement in a heterogeneous medium will vary in space depending on the posi- 

tion of the measurement. If a soil physical parameter such as the porosity is measured using a 

sample volume close to the actual value of a single particle or a pore, the measurement will 

vary dramatically between 0 and 100% depending on the position of the measurement (Figure 

4.1). If the sample volume (the scale) is increased, the fluctuations among repeated measure- 

ments will diminish. At this large scale, the measurement will be an average of all the micro- 

scopic variations in a continuous assembly of voids. The volume where a consistent popula- 

tion of data is obtained is defined as the representative elementary volume (REV, Bear, 1972). 

Different parameters may exhibit different spatial or temporal patterns, so that the REV for 

one parameter may differ from those for other parameters (Hillel, 1998). The measurement of 

soil permeability/saturated hydraulic conductivity on a structured soil is normally highly de- 

pendent on scale. 

4.2 Air and water permeability at two measurement scales 

All four papers in the current thesis deal with the issue of scale. Figures 4.2A to C show plots 

of ka measured on large (3140 cm 3 ) and small (100 cm 3 ) soil samples. In Figures 4.2A and B, 

measurements of ka related to the work in Paper I are shown. Here ka was measured at field 

water content on the large 3140-cm 3 soil samples under known boundary conditions (on-site 

measurements) in the A and B horizons of the studied soils. Three 100-cm 3 samples were, as 

explained earlier, extracted from each large ring and ka was measured on these samples as 

well. In general, the unstructured sandy soil (Jyndevad) showed no tendency of sampling 

scale whereas the more structured loamy soils (Fårdrup, Silstrup, and Slæggerup) showed a 

clear effect of scale, especially for measurements in the B horizon. The large differences in 

measurements of ka in the structured loamy soils using two different scales is probably related 

to preferential flow paths from plant root channels and animal burrows. When the sampling 

volume is increased the greater is the likelihood of intercepting larger, faster flow paths and, 

as a result, a larger value of ka is seen for the large soil cores. The small cores show a non- 

representative sampling of larger macropores and as a consequence, values of ka are generally 

lower. As seen in Figure 4.2B the effect of scale was most pronounced for the measurements 

in the B horizon, where the network of burrows was intact. In the disturbed and more ho- 

mogenous Ap horizon, the scale dependence effect was less pronounced although the majority 

39

Figure 4.1. Estimation of Representative Elementary Volume (REV) by increasing the bulk 

volume of soil VT to such an extent that porosity is independent of the position of the centre of 

VT (Kutílek and Nielsen, 1994). 

40

k a,lab [µm 2 ], 100 cm 3 

k a,lab [µm 2 ], 100 cm 3 

k a,lab [µm 2 ], 100 cm 3 

1000 

100 

10 

1000 

100 

10 

1000 

100 

10 

A 

B 

C 

Fårdrup 

Jyndevad 

Silstrup 

Slæggerup 

1:1 

Fårdrup 

Jyndevad 

Silstrup 

Slæggerup 

1:1 

Transect 1 

Transect 2 

1:1 

Higashi Hiroshima 

A horizon 

1 

1 10 100 1000 

k a,on-site [µm 2 ], 3140 cm 3 

A horizon 

B horizon 

Figure 4.2. Air permeability measured in the field on exhumed soil samples (ka,on-site) and in 

the laboratory on small 100-cm 3 soil samples (ka,lab). (A and B) Measurements on the 100-cm 3 

soils samples were carried out at the field water content (modified figure from Paper I). (C) 

Measurements on the 100-cm 3 soil samples were carried out at a controlled matric water po- 

tential of −100 cm H2O (modified figure from Paper IV). 

41

of the data points were still found below the 1:1 line. In the sandy, less structured soil at Jyn- 

devad, which did not show signs of macropores, sampling scale dependency was not apparent. 

The presence of macropores is probably also the explanation why the variability in ka within 

each set of three small rings was low for the unstructured Jyndevad soil, whereas it was high 

for the more structured soils. 

In Paper IV the same sampling technique with the same two sample sizes was used on 

a weakly structured sandy loam in an undisturbed constructed field in Japan. In the laboratory, 

ka was measured on 100-cm 3 soil samples at a controlled soil-water matric potential at −100 

cm H2O in contrast to the work in Paper I where ka was measured at field water content. 

Also here (Fig. 4.2C) an independency of scale was observed comparable with the unstruc- 

tured Jyndevad soil in Paper I. The difference between measurements on structured and un- 

structured soils is an illustrative example of the concept of a REV. Both the large and the 

small sample volumes are probably above the REV at Jyndevad, whereas the small ring sizes 

are apparently below the REV in the structured soils at Fårdrup, Silstrup, and Slæggerup, 

leading to a deviation between the two types of measurements. 

In Paper II the effect of scale was examined as well. Here kw and ka was measured at 

two scales (100 cm 3 and 6280 cm 3 ). The soil cores were sampled in four different Danish 

soils (Silstrup, Estrup, Jyndevad, and Tylstrup, Fig. 2.2) within three levels of the soil profiles 

corresponding to the A, B, and C/BC horizons. Air permeability was measured in the labora- 

tory on soil samples drained to a soil-water matric potential of −50 cm H2O. As in Paper I, an 

effect of scale was discovered. For measurements of ka and kw on the structured loamy soils 

(Silstrup and Estrup) the geometric means of large soil samples were generally lower com- 

pared to the geometric means of small soil samples (Figs 4.3A-D). Measurements of both ka 

and kw on the unstructured sandy soils (Tylstrup and Jyndevad, Figs. 4.3E-H) showed a much 

better agreement between scales of measurement. The large discrepancies between the soil 

measurements could probably also here be related to a non-representative sampling of larger 

pores in the small soil samples. 

The variation of macropores in some of the soil cores used in Paper II is probably also 

the explanation why in general the variability of the small rings was low for the unstructured 

soils compared to the more structured soils. A further examination of the variation between 

measurements revealed that the explanations for the phenomenon of variation between meas- 

urements were ambiguous. Variability in both ka and kw in the sandy soils was significantly 

higher for the large (6280 cm 3 ) samples compared to the small (100 cm 3 ) samples for half of 

42

100 cm 3 

k a,small [µm 2 ] 




100 

10 

1 

0.1 

0.01 

0.001 

100 

10 

1 

0.1 

0.01 

0.001 

100 

10 

1 

0.1 

0.01 

0.001 

100 

10 

1 

0.1 

Ap 

Bv 

BC(g)/C 

1:1 

Ap 

BE(g)/Bhs/Bt(g) 

C 

1:1 

Ap 

Bhs/Bs 

BC/C 

1:1 

Ap 

Bv/Ap2 

BC/C 

1:1 

Air Water 

Silstrup 

Estrup 

2D Graph 1 

0.01 

Tylstrup 

Tylstrup 

0.01 

0.001 

0.001 

0.001 0.01 0.1 1 10 100 0.001 0.01 0.1 1 10 100 

k a,large [µm 2 ] 

A 

Jyndevad 

C D 

E F 

G H 

6280 cm 3 

k w,large [µm 2 ] 

B 

Silstrup 

Estrup 

Jyndevad 

100 

10 

1 

0.1 

0.01 

0.001 

100 

10 

1 

0.1 

0.01 

0.001 

100 

10 

1 

0.1 

0.01 

0.001 

100 

Figure 4.3. Air permeability (ka,small and ka,large) at a matric water potential of −50 cm H2O 

and water permeability (kw,small and kw,large) measured on large (6280 cm 3 ) and small (100 

cm 3 ) soil samples. Values are geometric means. Error bars show ± one standard error (modi- 

fied figure from Paper II). 

43 

10 

1 

0.1 

k w,small [µm 2 ] 




100 cm 3

the individual horizons. Likewise, no clear effect in the variability was observed between the 

two sample sizes for the structured loamy soils. For the structured soils the variability be- 

tween measurements was lower for ka compared to kw. 

k a,small [µm 2 ], 100 cm 3 

100 

10 

1 

Ap 

Bv 

BC(g)/C 

1:1 

Silstrup 

0.1 

0.1 1 10 100 


A Ap 

Bhs/Bs 

BC/C 

1:1 

B 

3140 cm 3 /6280 cm 3 


Jyndevad 

0.1 1 10 100 

Figure 4.4. Air permeability (ka,small and ka,large) and water permeability (kw,small and 

kw,large) measured on large and small soil samples. Values are geometric means. Error 

bars show ± one standard error. Open symbols relate to Paper I where ka was measured at 

the field water content on 3140-cm 3 and 100-cm 3 soil samples. Closed symbols relate to Pa- 

per II where ka was measured at a water matric potential of −50 cm H2O on 6280-cm 3 and 

100-cm 3 soil samples. 

Figs. 4.4A and B show plots of ka measured at two scales at Silstrup and Jyndevad, 

which relates to the work of Papers I and II. Open symbols relate to Paper I where ka was 

measured at field water content on 3140-cm 3 and 100-cm 3 soil samples. Closed symbols relate 

to Paper II where ka was measured at a water matric potential of −50 cm H2O on 6280-cm 3 

and 100-cm 3 soil samples. For the soil at Silstrup there seemed to be a clear effect of scale 

between measurements on 100-cm 3 soil samples and the large soil sample (3140 cm 3 or 6280 

cm 3 ). For the soil at Jyndevad there seemed to be no effect of scale between measurements on 

the different sizes of soil samples. 

4.2.1 Scaling behaviour of permeability 

A method of modelling and identifying scale was proposed by Schulze-Makuch et al. (1999) 

who investigated the relation between kw (expressed through Ks) and scale of measurements. 

For heterogeneous media they found that Ks increased with scale of measurement to an upper 

44

oundary, after which the medium behaved as a homogenous medium and Ks remained con- 

stant. Up to the upper boundary, Schulze-Makuch et al. (1999) described the scaling behav- 

iour with the equation 

( ) m 

K = c V 

(11) 

s 

where c is the y-intercept of the regression line, V the volume of the tested material, and m the 

scaling exponent. Schulze-Makuch et al. (1999) found values of m between 0.45 and 0.55 

when measuring in heterogeneous porous flow media. Figure 4.5 is a plot of the ka measure- 

ment at Jyndevad and Silstrup relating to the three sizes of soil samples (100 cm 3 , 3140 cm 3 , 

and 6280 cm 3 ). As expected, a large effect of sample volume on ka (a high value of m) was 

seen for the structured soil at Silstrup whereas the unstructured soil at Jyndevad showed no 

such effect (value of m close to zero). That an upper boundary has been reached at the sam- 

pling volume of 3140 cm 3 for the soil at Silstrup, may reflect that the REV was reached for 

this sample size. Most likely it reflects that both samples had the same sample area (314 cm 2 ) 

although ka was measured on two different sample volumes (3140 cm 3 and 6280 cm 3 ). There- 

k a [µm 2 ] 

100 

10 

1 

0.1 

Silstrup 

Jyndevad 

10 -4 10 -3 10 -2 

Sample volume (m 3 ) 

Figure 4.5. Air permeability (ka) in relation to scale of measurement for the soils at Silstrup 

and Jyndevad. For the 100-cm 3 and the 6280-cm 3 soil samples ka was measured at a matric 

water potential of −50 cm H2O (Paper II). For the 3140-cm 3 soil samples ka was measured at 

the field water content (Paper I). Values are geometric means. Error bars show ± one stan- 

dard deviation. 

45

fore the two sample volumes gave more or less the same values of ka. Actually, there seemed 

to be a small decrease in the values of ka when the largest sample volume was reached. This 

could be because macropores are unlikely to be continuous through the entire soil sample 

when the length of the soil sample is doubled. 

4.2.2 Prediction saturated hydraulic conductivity at two scales 

The effect of scale on the ka- Ks relationship was examined as well. General log-log linear 

prediction relationships between ka and Ks for the studied soils in Paper II using the two 

sample sizes (100 cm 3 and 6280 cm 3 ) are presented in Figures 4.6A and B. Also plotted in the 

figures is the relationship established by Loll et al. (1999). The 95% prediction interval for 

the large samples (Fig. 4.6B) indicates an accuracy around ±1.4 orders of magnitude. The 

relationship of Loll et al. (1999) did not match the best-fit prediction relationship exactly, 

especially at the lower end of the scale where the lower 2.5% prediction line of Loll et al. 

(1999) is outside the lower 2.5% prediction line of the best-fit prediction relationship in the 

log (K s ) [m/d] 

4 

3 

2 

1 

0 

-1 

-2 

-3 

-4 

-5 

Small samples (100 cm 

-14 -13 -12 -11 -10 -9 

3 n = 171 

y = 1.29x + 14.55 

r 

) 

2 = 0.77 

A 

Large samples (6280 cm 

-14 -13 -12 -11 -10 -9 

3 n = 59 

y = 0.94x + 10.90 

r 

) 

2 = 0.45 

B 

Silstrup 

Estrup 

Jyndevad 

log(k a ) [m 2 ] 

log(k a ) [m 2 ] 





Figure 4.6. General log-log linear prediction relationship between ka (measured at a matric 

water potential of −50 cm H2O) and Ks measured on 100-cm 3 and 6280-cm 3 samples for all 

soils and horizons combined in Paper II. The 95% prediction interval, n, α, β, and r 2 are 

given. Also plotted is the relationship and 95% prediction interval between ka (measured at a 

matric water potential of −100 cm H2O) and Ks found by Loll et al. (1999) on 100-cm 3 sam- 

ples (figure from Paper II). 

46

present study. This is probably explained by the few numbers of measurements carried out at 

the lower end of the scale. At the higher end of the scale, the agreement between the two rela- 

tionships was better. Statistical analyses using the BCa percentile boot-strapping method re- 

vealed that α for the large samples had a value that was significantly lower (but only margin- 

ally) compared to the relationship of Loll et al. (1999). The average value of log(Ks) found 

from the general prediction relationship in this study was not significantly different to that 

found by Loll et al. (1999). The results in the present study indicate that the Ks-ka relation- 

ship can be used on both large and small soil samples without any larger misinterpretation of 

the results. 

4.3 Summary 

When increasing the sample volume (the scale), the fluctuations among repeated measure- 

ments will diminish. On a large scale, the measurement will be an average of all the micro- 

scopic variation in a continuous assembly of voids. The volume where a consistent population 

of data is obtained is defined as the representative elementary volume (REV). The measure- 

ment of soil permeability on a structured soil is normally highly dependent on scale. 

When increasing the sample volume from 100 cm 3 to 3140 cm 3 /6280 cm 3 , large differ- 

ences in the values of ka and kw were discovered when measuring in structured soils. This 

was probably related to a non-representative sampling of plant root channels and animal bur- 

rows. When the sampling volume was increased, the likelihood of intercepting larger, faster 

flow paths increased and, as a result, a larger value of ka was seen for the large soil cores. 

When measuring in a sandy, less structured soil, a sampling scale dependency was not dis- 

covered. The presence or absence of macropores in some of the soil cores may probably also 

explain why the variability of the small rings in general was low for the unstructured soils 

compared to the more structured soils. 

The effect of scale on the ka-Ks relationship was examined as well when measuring on 

100-cm 3 and 6280-cm 3 soil samples. Here, the results indicated that compared with the gen- 

eral prediction relationship of Loll et al. (1999), the Ks-ka relationship can be used on both 

large and small soil samples without any larger misinterpretation of the results. 

47

5 Spatial variability 

As explained earlier, variability can be partitioned into two broad classes, random and sys- 

tematic. The regionalised variable theory assumes that the spatial variation of any variable can 

be expressed as the sum of three major components (Burrough and McDonnel, 1998): A 

structural component having a constant mean or trend, a random, but spatially correlated 

component, known as the variation of the regionalised variable, and a spatially uncorrelated 

random noise or residual error term. The basic tool of geostatistics is the semivariogram. The 

semivariance function (Journel and Huijbregts, 1978) is defined as 

γ ( h) = (½)Var[ Z( x) − Z( x+ h)] 

(12) 

where Z(x) and Z(x+h) are the values of a soil property Z at location x and x + h, respectively, 

h being the distance separating the two values, and Var[Z(x) - Z(x + h)] is the variance of the 

difference between the values of the soil properties. Assuming that the mean of the random 

function Z(x) is stationary and that the variance of the differences between sample values is 

finite and depends only on h, the semivariance is estimated 

* 1 n 

γ ( h) = ∑ [ Z( x ) −Z( 

x )] 

2n i i 

i = 1 

2 

+ h (13) 

where γ*(h) is the sample (experimental) semivariance and n is the number of pairs of data 

points separated by the distance h. A plot of a classical experimental semivariogram (Fig. 

1.2C) is described by three parameters: (i) the sill, (ii) the range, and (iii) the nugget. The 

horizontal part of the semivariogram curve, the sill, shows the values of h where there is no 

spatial dependence between the data points. The semivariance value at the sill corresponds to 

the total variance of the system. The curve rise from h=0 to the sill is called the range. The h 

value at the end of the curve rise is an estimate of the distance between points at which sam- 

ple values become independent. The intercept at h=0, known as the nugget, is an expression 

of the variance of measurement errors combined with that from spatial variation at distances 

shorter than the sample spacing. The interpretation of the spatial structure of the variable of 

interest is accomplished by modelling the semivariogram. Commonly used models are linear, 

power functions, spherical, exponential, Gaussian, or cubic models (Upchurch and Edmonds, 

1991). 

The goal of the study of spatial variability is most often to estimate the value of the 

regionalised variable at points that have not been visited. The process of interpolation be- 

tween sampled points using the spatial structure described by the semivariogram is named 

kriging. Kriging yields more realistic estimates than older methods of linear interpolation, as 

49

it considers the spatial trend of property based on the array of values surrounding the point of 

interests. The kriging system provides both the estimate and a value for the error associated 

with that estimate. 

Since ka both in the field and in the laboratory is simpler and faster to measure than 

kw, it provides an easier way of characterising undisturbed soil air and water transport proper- 

ties as well as the soil pore size distribution and aggregation. The current study (Papers III 

and IV) therefore aimed at investigating if measurements of ka could be an efficient tool to 

describe the spatial variability of the infiltration parameter of the soil. 

5.1 Geostatistical analysis of air permeability 

In Paper III, ka,in situ was measured in the topsoil in two small agricultural catchments (Ans 

Field Slope and Rødding Field Slope) in a 30-m grid using the portable air permeameter. Air 

permeability was measured in situ in 43 grid points at Ans Field Slope and in 29 grid points at 

Rødding Field Slope. In addition, four extra measurements at each point were carried out 

symmetrically around three grid points at each site in a radius of five meters in order to exam- 

ine the short distance variability (Fig. 5.1). The estimated semivariograms of the measured 

values of log(ka,in situ) at both Ans and Rødding Field Slopes are shown in Figures 5.2A and B. 

Both semivariograms were fitted using a spherical model. The plotted data from Ans (Fig. 

5.2A) showed some scattering at high lags but showed also a clear range at approximately 110 

m. Data from Rødding (Fig. 5.2B) showed a high degree of scatter at high lags. Here, data 

showed a spatial dependency over a distance of approximately 90 m. 

On the Japanese soil in Paper IV, ka was measured on-site (with known boundary 

conditions) in the undisturbed constructed field along two 70-m-long transects using the port- 

able air permeameter presented in Paper I. The two transects were laid out perpendicular to 

the tillage direction and each of them was divided into 36 sampling points 2 m apart. Figure 

5.2C shows the estimated semivariograms for ka,on-site measured along the two transects. For 

the Japanese soil the ka,on-site measurements indicate a spatial dependency over a distance of 18 

m and 32 m for the two transects, respectively. The results from the two studies (Papers III 

and IV) therefore indicate that it is possible to explain the spatial structure of ka using the 

developed portable air permeameter. 

50

Figure 5.1. Outline of Ans and Rødding Field Slope showing the placement of the grid points 

used for the ka,in situ measurements. 

51

Semivarians(h) 



0.30 

0.25 

0.20 

0.15 

0.10 

0.05 

0.00 

0.30 

0.25 

0.20 

0.15 

0.10 

0.05 

log k a,in situ 

log k a,in situ 

Ans 

Rødding 

0.00 

0 

0.30 

50 

log ka,on-site 100 150 200 

0.25 

0.20 

0.15 

0.10 

0.05 

Transect 1 

Transect 2 

0.00 

0 10 20 30 40 50 60 

h [m] 

A 

B 

C 

Higashi-Hiroshima 

Figure 5.2. (A and B) Estimated semi-variograms for log(ka,in situ) measured at Rødding and 

Ans Field Slope (modified figures from Paper III). (C) Estimated semi-variogram log(ka,on-site) 

measured in a constructed field in Japan (recalculated data from Paper IV). All semi- 

variograms are fitted using a spherical model. 

52

5.2 Spatial variation of air permeability at two field slopes 

The back transformed kriged values of ka,in situ for both field slopes are shown in Figures 5.3A 

and B where the values are draped over a digital terrain model of the two field slopes. For 

Ans Field Slope (Fig. 5.3A, pixel size equal to 3 m), values of ka,in situ varied about one order 

of magnitude. Low values of ka,in situ were associated with the steepest part of the field slope 

near the outlet. High values of ka,in situ were associated with the less steep parts of the field 

slope at the top of the slope. The steepest parts also had the highest content of coarse sand in 

contrast to the less steep parts with a low content of sand. The difference in the sand content 

(and clay content) on the field slope was probably explained by an erosion of the finer parti- 

cles at the steepest part of the slope in connection with surface runoff events. The part with 

the lowest content of sand also had the strongest structure with a large number of macropores. 

Therefore the low values of ka,in situ on the steep part of the slope were probably related to the 

weak structure and low number of macropores. At Rødding Field Slope, values of ka,in situ also 

varied by one order of magnitude (Fig. 5.3B, pixel size equal to 2 m), but values were gener- 

ally lower compared to Ans Field Slope. At Rødding Field Slope no clear relation between 

variability of ka,in situ and texture or steepness was seen. 

5.3 Summary 

The basic tool of geostatistics is the semivariogram, which can be used to describe the spatial 

dependency among data points in an area. Since ka both in the field and in the laboratory is 

simpler and faster to measure than kw, it provides an easier way of characterising undisturbed 

soil air and water transport. The aim of the current study was therefore to investigate if meas- 

urements of ka could efficiently describe the spatial variability of the infiltration parameter of 

the soil. 

In the current study, ka,in situ was measured in the topsoil in a 30-m grid in two small 

agricultural catchments using the developed portable air permeameter. The estimated semi- 

variograms showed a spatial correlation of the log transformed values with a range of ap- 

proximately 100 m. On-site air permeability measurements (known boundary conditions) in 

an undisturbed constructed field in Japan indicated a spatial dependency of the log trans- 

formed data of approximately 20 m. The results from the studies therefore indicate that spatial 

structure could be efficiently and reliably described by ka measurements. This opens up for 

new methods of characterising the soil while obtaining new and valuable information about 

soil variability. 

53

Figure 5.3. Digital terrain models of Ans and Rødding Field Slopes. Overlays are kriged val- 

ues of ka (µm 2 ) measured in situ (ka,in situ). Pixel size for Ans Field Slope is 3 m. For Rødding 

Field Slope pixel size is 2 m (figure from Paper III). 

54

6 Modelling surface runoff 

One of the most sensitive parameters in hydrological modelling including surface runoff 

models is Ks (Rudra et al., 1985; De Roo and Riezbos, 1992; Fisher et al., 1997; De Roo and 

Jetten, 1999; Jetten et al., 1999). As discussed earlier, the variability of Ks is typically high 

and the measuring methods are time-consuming. Therefore accurate and fast methods of 

measuring the hydraulic characteristics of the soil are needed to obtain a detailed knowledge 

of the spatial variability. It seems likely that ka,in situ measured near field capacity could be a 

good prediction of the permeability of the entire pore system and thereby a good prediction of 

Ks. Therefore measurements of ka could be a substitute for the more time-consuming meas- 

urements of Ks in order to get a detailed picture of the spatial variability of the infiltration 

parameter in a small catchment. 

6.1 Key parameters and processes in relation to surface runoff 

The water discharge measured at the downstream boundary of a catchment is often separated 

into three different forms: groundwater flow, subsurface flow, and surface runoff. Groundwa- 

ter flow is the base flow component of the discharge and is relatively constant through time. 

Subsurface flow and surface runoff are mostly generated in connection with individual pre- 

cipitation events. Surface runoff is generated when the water supply rate (rain, irrigation, or 

snowmelt) to the soil surface exceeds the infiltration rate of the soil. In connection with high 

intensity or long duration rainfall events the surface layer can become completely saturated 

and if the surface is not completely horizontal, the surplus water starts to run downslope as 

surface runoff. In Denmark, surface runoff (and erosion) is mainly associated with agricul- 

tural land. Earlier studies have pointed out that surface runoff and erosion is an increasing 

problem in Denmark (e.g. Schjønning et al., 1995). The major concern when considering sur- 

face runoff is the increased risk of leaching of nutrients to watercourses and subsequent eu- 

trophication. 

Besides the infiltrability of the soil, several factors control the generation and amount 

of surface runoff, such as: slope gradient, slope length and shape, surface roughness, plant 

cover, and the length and intensity of the precipitation event. Also freezing of the soil can 

result in reductions in the infiltration rate, which increases the potential for surface runoff. 

55

Normally the generation of surface runoff can be described by two different mecha- 

nisms: Infiltration excess runoff (IER) or saturation excess runoff (SER, Freeze, 1980; Coles 

et al., 1997). 

Figure 6.1. Mechanisms of surface runoff. Moisture content versus depth profiles for (a) the 

IER mechanism and (b) the SER mechanism. Surface runoff generation for (c) the IER 

mechanism and (d) the SER mechanism (Freeze, 1980). 

The classic runoff mechanism, IER, happens when the rainfall intensity exceeds Ks of 

the surface soil. The moisture content at the surface will then increase as a function of time 

(Fig. 6.1A). At some point in time (t 3 in Fig. 6.1A) the surface becomes saturated and an in- 

verted zone of saturation begins to propagate downward into the soil. At this time (Fig. 6.1C) 

the infiltration rate drops below the rainfall rate and surface runoff is generated. The time t 3 is 

known as the time of ponding. 

56

The SER mechanism is caused by a rising water table, which saturates the surface 

from below (Figs. 6.1B and D). In this case the rainfall intensity is lower than Ks and ponding 

as well as surface runoff occurs when no further soil moisture storage is available. 

IER is more common on areas upslope where Ks is low whereas SER is more common 

at the base of hillslopes (Freeze, 1980). SER also takes place on soils where a relatively per- 

meable topsoil layer overlies less permeable material (e.g. a ploughpan, an argillic horizon, or 

a fragipan). 

When modelling surface runoff, the nature of the flow is often described by the kine- 

matic wave approximation of the Saint-Venant equations (Chow et al., 1988; Giráldez and 

Woolhiser, 1996). By integrating both the mass conservation equation and the dynamic con- 

servation equation, it is possible to find an analytical or a numerical solution depending on the 

type of input (rainfall pattern) and representation of watershed geometry (Gerits et al., 1990). 

The mass equation states that the combination of the time variation of water depth h, at 

any point, and the change of flow rate q, with distance x, equals the excess of rainfall 

∂h ∂q 

+ = r − f 

∂t ∂x 

(14) 

where t stands for time and r and f for rainfall and infiltration rates, respectively. 

One simplification of the dynamic conservation equation is the relation for uniform 

flow between flow rate and water depth. 

q = α h 

m 

(15) 

where α is a coefficient expressing surface conditions for the flow and m is an empirical de- 

termined exponent. 

For a infinitely wide channel without sides represented by a laterally uniform sloping 

surface with a layer of water flowing over it, q may be expressed as 

S 

q h 

n 

0 5/3 

= (16) 

where S0 is the slope and n is Manning's n. Eq. (16) is therefore also known as Manning's 

equation. 

6.2 LImburg Soil Erosion Model (LISEM) 

In Paper III, the proposed method of using measurements of ka,in situ to characterise spatial 

variability in Ks is illustrated with the application of a distributed surface runoff model to two 

57

small agricultural catchments. The distributed surface runoff model named LImburg Soil Ero- 

sion Model (LISEM, De Roo et al., 1996b) 1 was used for the simulations. LISEM is a physi- 

cally-based hydrological and soil erosion model completely integrated into a GIS (Van Deur- 

sen and Wesseling, 1992). The main reason for using a GIS is that runoff and soil erosion 

processes vary spatially, so that cells should be of a size to allow for spatial variation. Also, 

the data for the large number of cells required are enormous and cannot easily be entered by 

hand, but can be obtained with a GIS. GIS can compute maps of altitude, slope, and aspect, 

which are all input for the LISEM model. Because detailed field sampling of input variables is 

not feasible, a limited number of point observations of the soil, e.g. collected during field ex- 

periments, are often available. Geostatistical interpolation techniques, incorporated in the 

GIS, can be used to produce maps from these point observations. 

LAI, Cov 

Ksat, theta 

ldd 

n, slope 

RR 

Rainfall 

INTERCEPTION 

INFILTRATION 

SURFACE 

STORAGE 

OVERLAND 

FLOW 

Water 

Discharge 

SPLASH 

EROSION 

FLOW 

EROSION 

TRANSPORT 

DEPOSITION 

Sediment 

Dischange 

Cov, AS 

COH 

water flux 

sediment flux 

control link 

D50 

input var 

Processes 

calculated 

within a grid cell 

Kinematic Wave 

for transport 

between cells 

Processes repeated 

surface and 

channel cells 

LAI – Leaf Area Index n – Manning's n 

Cov – Fraction of soil covered by vegetation AS – Aggregate stability 

Ksat – Saturated hydraulic conductivity COH– Cohesion of soil 

theta – soil water content D50 – Median grain size 

RR – Random Roughness 

Figure 6.2. Flowchart of LISEM including the main processes. 

1 See also the LISEM website (http://www.geog.uu.nl/lisem/) 

58

LISEM is able to incorporate processes such as interception, surface storage in micro- 

depressions, infiltration, vertical movement of soil water, surface runoff, channel flow, splash 

and flow detachment. The model can be used for research, planning, and conservation pur- 

poses in hydrological catchments and is built to simulate both the effects of the current land 

use and the effects of soil conservation measures. A flowchart of LISEM is shown in Figure 

6.2. After rainfall begins, the vegetation canopy intercepts some of the precipitation until the 

maximum interception storage capacity has been met. Besides interception, direct throughfall 

and leaf drainage occur, which, together with overland flow from upslope areas, contribute to 

the amount of water available for infiltration. Excess water accumulates on the surface in mi- 

cro-depressions. When a predefined depression volume has been filled, overland flow begins. 

In LISEM, surface runoff is calculated using Manning's n and slope gradient, with a direction 

according to the aspect of the slope. For the distributed surface runoff routing, a four-point 

finite-difference solution of the kinematic wave is used together with Manning's equation 

(Chow et al., 1988). The solution of the kinematic wave is done over a “local drain direction 

map” that forms a network connecting each cell into eight directions. When rainfall ceases, 

infiltration continues until depression storage water is no longer available. Either raindrop 

impact or overland flow can cause both soil detachment and transport. Infiltration in LISEM 

can be calculated in several ways. This study used either the one or the two layer Green and 

Ampt approach which needs input of Ks, pore volume, initial moisture status, and the soil 

water tension at the wetting front. Soil water tension at the wetting front was estimated fol- 

lowing the outline of Rawls et al. (1983) using the Brooks-Corey constants (Brooks and 

Corey, 1964) found from the soil water retention data (Brakensiek, 1977). The Green and 

Ampt approach is a simplistic, but still useful, theoretical approach for modelling the infiltra- 

tion into a soil. The main assumptions of the Green and Ampt approach are that a distinct and 

precisely definable wetting front exists during infiltration, and that, although this wetting front 

moves progressively downward, it is characterised by a constant matric suction, regardless of 

time and position. 

A distributed model such as LISEM attempts to increase the accuracy of the resulting 

simulation by preserving and utilising information concerning the areal distribution of all spa- 

tially variable processes incorporated into the model (De Roo et al., 1989). As models get 

increasingly realistic it will need more and better data. A sensitivity analysis of an earlier ver- 

sion of LISEM showed that most critical parameters in the model seemed to be the spatial and 

temporal variability of the soil hydraulic conductivity and the initial water content (De Roo et 

al., 1996a). Often, only a limited number of field measurements of these two variables are 

59

available, which can have a large consequence on the simulation result. A high number of 

input maps in the model are therefore created form a limited amount of field data and are 

therefore subjective which might lead to erroneously outputs from the model. Beside this, the 

infiltration process in LISEM is 1-dimensional contrary to most other processes in the model 

which are 3-dimensional. This means that a process as subsurface flow is not included in the 

model. However, results of Ritsema et al. (1996) indicated that such processes are not impor- 

tant at storm level. 

6.3 Modelling surface runoff at two field slopes 

Ideally, any test of the effect of scale and spatial variability of the infiltration parameter on 

calculating runoff in a catchment requires knowledge of the actual runoff and erosion and a 

complete set of data for the related variables. In the two examined catchments (Ans and Rød- 

ding Field Slopes), runoff was measured in the winter half-year during a period of four years. 

Several runoff events were measured with most of them occurring when the soil was frozen or 

partially frozen. Only a few runoff events occurred during wintertime when the soil was com- 

pletely unfrozen. Figure 6.3 is an example of a runoff event, which occurred at 

Precipitation [mm/h] 

2.0 

1.5 

1.0 

0.5 

0.0 

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 

Figure 6.3. Measured surface runoff event at Ans Field Slope, December 25 th , 2000. 

hour 

Precipitation 

Surface runoff 

Ans Field Slope, on 25 December, 2000. During a period of 13 hours, 22 mm of low intensity 

rain fell over the catchment. Runoff of the SER type began after one and a half hour on a 

60 

18 

16 

14 

12 

10 

8 

6 

4 

2 

0 

Surface runoff [l/s]

completely unfrozen soil. Since the generation of runoff of the SER type is dominated by a 

limitation of the soil moisture storage in the soil, it is only to a minor extent controlled by Ks. 

Also such a long-duration storm event will probably need an infiltration module including 

subsurface flow. It was therefore decided not to test the effect of scale and spatial variability 

on these kinds of event. Instead, a “virtual” event representing a summer high-intensity rain- 

fall event producing surface runoff of the IER type was chosen for simulation. Since only a 

small amount of the total precipitation normally contributes to surface runoff during this kind 

of event, the effect of changing the distribution of the infiltration parameter will be larger 

compared to events of the SER type. Surface runoff modelling in Paper III only focused on 

the spatial variability of Ks. All other parameters were given a constant value over the entire 

field corresponding to a typical summer situation of a winter wheat field. Also, all parameters 

except Ks were given the same values between each simulation. For the Ans Field Slope the 

size of the pixels (cells) was 3 m. For the Rødding Field Slope the size of the pixels was 2 m. 

For each scenario, different sub-scenarios of ka were constructed by kriging with six different 

resolutions. For the Ans Field Slope resolutions of Ks of 3, 9, 15, 30, 60 and 90 m were used 

as input to the model. For the Rødding Field Slope resolutions of Ks of 2, 6, 10, 20, 40, and 

60 m were used as input. Additionally, a reference-scenario for both field slopes was set up 

using the geometric average of ka for the whole catchment. All GIS maps used for the simula- 

tions had the same pixel sizes. Simulations were carried out using a design rainstorm with a 

total duration of 60 minutes. 

The study of Paper II showed that although values of ka on structured soils depended 

highly on the sample size, the general log(Ks)–log(ka) prediction relationship of Loll et al. 

(1999) could be applied on both large and small soil samples without any larger misinterpreta- 

tion of the results. Scenarios were therefore constructed using values of Ks estimated from the 

site-specific log-log prediction relationship presented in Eq. (10). In order to test the effect of 

different relationships on the resulting runoff, five different values of α and β derived from 

BCa percentile bootstrap estimates were chosen. Also tested was the performance of the gen- 

eral prediction relationship of Loll et al. (1999), Eq. (8). The six different relationships com- 

bined with the seven different resolutions of Ks gave a total of 42 LISEM simulations for each 

field slope. 

The simulations resulted in surface runoff of the IER type. Figure 6.4 shows the peak 

height and the total runoff for the 42 different sub-scenarios at Ans and Rødding Field Slope. 

To obtain a non-zero output of surface runoff during the simulations, an adjustment had to be 

61

made to Ks (factor 0.1). This made it possible to compare surface runoff outputs for every 

combination of scenario. 

The Green and Ampt model is sensitive to the choice of Ks and initial moisture con- 

tent. The initial assumption that the wetting front moves down as a wet body parallel to the 

surface with a speed dictated by the Ks is not correct. Many researchers therefore have sug- 

gested “field” variables: a “field porosity” or a “field Ks”, or a suction at the wetting front that 

is not the matrix po tential at a given time but is more a soil property. This means that calibra- 

tion of surface runoff models is a normal procedure when attempting to model actual surface 

runoff events. Jetten et al. (1999) made overall adjustments of Ks ranging from a factor of 

0.15 to 0.30 when evaluating a range of different field-scale and catchment scale soil ero- 

Peak [l/s] 

Total runoff [mm] 

300 

250 

200 

150 

100 

50 

0 

10 

5 

0 

Size of sub-area [m]: 

Av. 90 60 30 15 9 3 

1 10 100 1000 10000 

Number of sub-areas 

α = 0.83 

α = 1.11 



A Size of sub-area [m]: 

B 

Av. 60 40 20 10 6 2 

α = 1.38 

α = 1.64 

Rødding Field Slope 

C D 

Rødding Field Slope 

1 10 100 1000 10000 

Number of sub-areas 

α = 1.90 

α = 1.27 (Loll et al., 1999) 

Figure 6.4. Simulated peak height and total runoff volume for Ans and Rødding Field Slopes 

in relation to the resolution of model input of Ks using the Green and Ampt infiltration equa- 

tion. The output of the model was tested using five different log-log linear relationships be- 

tween ka and Ks found from the bootstrap estimates in Paper II. Also tested was the relation- 

ship of Loll et al. (1999). Six different resolutions of kriged Ks values were chosen for the two 

catchments. Also tested was a reference scenario where the geometric average of Ks was used 

(“Av”; corresponding to 1 sub-area), figure from Paper III. 

62

sion models. In this context they concluded that uncalibrated use of catchment models is not 

advisable. Calibration is imperative for small and medium scale catchment applications, 

where the effects of spatial variability on the runoff and erosion processes strongly influence 

the simulation. 

By increasing the resolution of Ks there seemed to be an upper limit for both field 

slopes where the peak height and the amount of runoff were independent of resolution scale 

(Fig. 6.4). For the Ans Field Slope this upper limit resolution was c. 30 m corresponding to 31 

sub-areas for the whole catchment. For the Rødding Field Slope the upper limit was c. 40 m 

corresponding to 10 sub-areas for the whole catchment. For low values of α (and correspond- 

ingly low values of β) the effect of resolution scale was less pronounced, due to low values of 

Ks and a relatively uniform distribution. The high discharge rates then meant that every part 

of the catchment was producing surface runoff and the effect of resolution scale then became 

less important. 

The effect of changing the resolution with a spatially variable pattern of a parameter 

value (here Ks) is similar to the concept of a representative elementary area (REA) introduced 

by Wood et al. (1988). According to the REA concept, at a certain scale the hydrological re- 

sponse becomes invariant or varies only slowly. The concept is comparable with the REV 

concept presented earlier. The REA concept is a way of simplifying the catchment since it 

promises a scale over which the process representations remain simple. Wood et al. (1988) 

analysed the effect of scale by dividing the catchment into smaller subcatchments and deter- 

mined the average water fluxes for each subcatchment, in contrast to this work where the total 

output from the catchment is considered. However, if the REA concept of Wood et al. (1988) 

is applied to the present study it can be concluded that the Ans Field Slope had an REA of 30 

× 30 m 2 and Rødding Field Slope had an REA of 40 × 40 m 2 . If an REA exists for the two 

field slopes, the size of the sampling grid seems to be adequate for both of them. The same 

can be concluded when taking the result of the semivariograms into account. Apparently, 

nothing is gained by choosing a resolution of Ks that is higher than the found REAs of the two 

catchments for the given simulations. 

Figure 6.5 shows selected hydrographs from the results of the simulations for Ans 

Field Slope. The hydrographs represent outputs from the model using values of Ks estimated 

from the site specific log(ka)-log(Ks) linear relationship found in Paper III (α=1.38). Sub- 

scenarios using resolutions of ka of 3, 60 and 90 m and a fourth scenario using the geometric 

63

Runoff [l/s] 

120 

100 

80 

60 

40 

20 

0 

6 mm 

7 mm 


13 mm 

7 mm 

0 20 40 60 

Time [minutes] 

6 mm 

Precipitation 

3 m (3117) 

60 m (8) 

90 m (3) 

Figure 6.5. Examples of simulated hydrographs for Ans Field Slope using the site specific 

log-log linear relationship between ka and Ks found in this work (α=1.38). Hydrographs 

show simulation using three different resolutions (number of sub-areas) of ka (figure from 

Paper III). 

average of ka were selected. From Figure 6.5 it is clear that the introduction of spatial vari- 

ability into the model had a significant effect on the output. Note that data of the geometric 

average of ka,in situ is not plotted since no runoff was produced when running the simulation 

with this scenario. Using the kriged values of ka,in situ (resolution of 3 m) the amount of surface 

runoff had a significant peak height of 30 l/s and a total amount of surface runoff at 0.7 mm 

corresponding to a discharge rate (total amount of runoff divided with total amount of precipi- 

tation) of 1.8%. The highest output from the model was obtained with the kriged values of 

ka,in situ using a resolution of 90 meters. With the kriged values as input to the model, parts of 

the catchment were given low values and parts were given high values compared to the geo- 

metric average. The lowering of ka,in situ at some parts of the catchment resulted in a general 

rise in surface runoff. This example illustrates the importance of incorporating the spatial 

variability of the hydraulic conductivity into surface runoff models and the ability of using a 

distributed runoff model. However, as indicated in Figure 6.4, spatial trends in Ks and the 

corresponding effect on peak height and runoff volume is highly dependent on the rate of sur- 

face runoff compared to the amount of precipitation. This has also been pointed out by several 

64 

80

authors (Smith and Hebbert, 1979; Ogden and Julien, 1993; Woolhiser et al., 1996; Merz and 

Plate, 1997; Merz and Bardossy, 1998). 

6.3.1 Many measurements of air permeability versus few measurements of saturated 

hydraulic conductivity 

Using a given amount of measurement time in the field, measurements of ka have the advan- 

tage that a lot of points in the field can be visited compared to the more time-consuming tradi- 

tional measurement of Ks where only a few points at the field can be visited. This raises the 

question what the differences in the output from the model would be when applying one of 

the two types of measurements into the model. In order to test this, twenty scenarios at the 

Ans Field Slope were constructed in Paper III dividing the catchment into six sub-areas (90- 

meter resolution). From the grid points within each sub-area a random value of ka,in situ was 

chosen. The scenarios were supposed to represent cases were Ks was estimated using a tradi- 

tional measurement method then only being able to carry out a few measurements over the 

field. Values of ka were converted to values of Ks by means of the site specific prediction 

relationship found in this work (α=1.38, β=15.13). The modelling simulations of the twenty 

scenarios are shown in Figure 6.6 together with the two model simulations using the kriged 

values of ka with a resolution of 3 meters. This to represent scenarios where Ks was estimated 

using the general prediction relationship of Loll et al. (1999) or the site specific prediction 

relationship found in the current work. If only a few randomly chosen values of Ks were used 

to represent the spatial variation within the field slope, large deviations in repeated simulation 

results were obtained, both with respect to peak height (0 to 158 l/s) and total runoff. When 

using many estimated Ks values (from ka,in situ measured in every grid point) based on a Ks-ka 

relation, the model generally showed relatively comparable outputs (17 and 30 l/s, respec- 

tively) in contrast to the scenarios using traditional measurements of Ks. 

Uncertainties in the output from the model can be related to a number of causes. Un- 

certainties in the measurement of ka,in situ exists not only in relation to the uncertainty of the 

lower border conditions in relation to the in situ measurement (the shape factor), but also to 

the difficulties describing the spatial variability of the infiltration parameter in the catchment 

correctly. Uncertainties are also related to the determination of the prediction relationship 

between ka and Ks hereunder the assumption that the same relation exists on both large and 

small soil samples. When upscaling a point measurement (such as ka,in situ) to a larger scale, 

effects such as the randomness of the spatially correlated infiltration parameter can be diffi- 

cult to interpret. Since Ks is one of the most sensitive parameters in hydrological modelling 

65

Runoff [l/s] 

200 

150 

100 

50 

0 

6 mm 

7 mm 

Max for random 

simulations 

Min for random 

simulations 

13 mm 

7 mm 

6 mm 

0 20 40 60 80 

Time [minutes] 

Precipitation 

Random 

log(K s )=1.27 log(k a ) + 14.11 

(Loll et al., 1999) 

log(K s )=1.38 log(k a ) + 15.13 

(present study) 


Figure 6.6. Simulated hydrographs at Ans Field Slope using the Green and Ampt infiltration 

equation. For the simulations using random values, the field slope was divided into 6 sub- 

areas and a random value of ka measured in situ was chosen within each sub-area. Also 

shown is the simulation outputs using the general relationship of Loll et al. (1999) and the 

site specific relationship found in this work (α=1.38) using a spatial resolution of ka at 3 m 

(3117 sub-areas), figure from Paper III. 

the choice of an infiltration model will always be important especially when simulating sur- 

face runoff of the IER type. Here, the prevailing part of the total precipitation is infiltrated 

into the soil whereas only a small part is responsible for the surface runoff. Therefore even 

relatively small changes in the total infiltration will have a large effect on the amount of sur- 

face runoff leading to uncertainties in the output. Also, infiltration models often have prob- 

lems dealing with the hydraulic conductivity at near-saturation. As exemplified in Figure 6.7 

large uncertainties exist between the near-saturated and the saturated hydraulic conductivity 

where the conductivity might vary op to more than an order of magnitude for a structured 

loamy soil. This will lead to uncertainties in the output from model especially when using a 

rather simple infiltration model such as the Green and Ampt model. 

66

K(h) [cm/d] 

100 

10 

1 

0.1 

K s 

K s 

10 100 

Matric water potential [|cm|] 

Figure 6.7. Example of values of Ks (square symbol, geometric mean of two soil samples) and 

unsaturated hydraulic conductivity (triangle and circle symbol) measured on a loamy soil 

(Lindhardt et al., 2001). The solid line is a fit of the unsaturated hydraulic conductivity using 

the Mualem-Brooks & Corey k(h) model. 

6.4 Recommendation for using the proposed Ks-ka approach in runoff modelling 

Given the large spatial variation of most soil hydraulic properties, cheap information and 

many less precise measurements can often be more efficient than a few more expensive and 

precise measurements (Minasny and McBratney, 2002). Since massive measurement efforts 

will normally be required to get a satisfactorily representation of the spatial variability in Ks, 

the use of ka,in situ to assess spatial variability in Ks appears a promising alternative. However, 

some recommendations can be given to minimise the sources of errors. When measuring ka it 

is important assuring that the soil is sufficiently drained. As exemplified in this thesis some 

soils having a high frequency of medium-sized pores might not always be sufficiently drained 

even at field capacity leading to a high blockage of the pores in the soil when measuring ka. 

In these kinds of situations a useful relationship might be difficult to establish. Another im- 

portant issue is the design of the measurement in the field in order to establish a correct pic- 

ture of the spatial variability in the catchment. Even though it appeared that in this work the 

chosen grid resolution in the field was sufficient for an analysis of the spatial structure of ka 

this might be different for other areas. Finally, it should be emphasised that regardless of 

67

choice of model for simulating infiltration and surface runoff, the sensitivity of the model 

output to Ks (ka) will often be large and there will be a need for calibrating the model when 

simulating actual runoff events. 

Fejl! Ukendt argument for parameter.6.5 Summary 

One of the most sensitive parameters in hydrological models including surface runoff models 

is Ks. Measurements of ka can be a substitute for the few, more time-consuming measure- 

ments of Ks in order to get a detailed picture of the spatial variability of the infiltration pa- 

rameter in a small catchment. Surface runoff is generated when the water supply rate to the 

soil surface exceeds the infiltration rate of the soil. In the current work, a distributed surface 

runoff model was used for the simulations. A site-specific Ks-ka prediction relationship pre- 

sented earlier was applied in the surface runoff model. The simulation results were highly 

dependent on whether the geometric average or kriged values of ka were used as model input. 

If only a few randomly chosen values of Ks were used to represent the spatial variation within 

the field slope, large deviations in repeated simulation results were obtained, both with re- 

spect to peak height and hydrograph shape. On the other hand, when using many estimated Ks 

values (from ka,in situ measured in every grid point) based on a Ks-ka relation, the model gener- 

ally showed relatively comparable outputs in contrast to the scenarios using traditional meas- 

urements of Ks. Since massive measurement efforts will normally be required to get a satis- 

factory representation of the spatial variability in Ks, the use of ka,in situ to assess spatial vari- 

ability in Ks appears to be a promising alternative. Recommendations for using the proposed 

Ks-ka approach in runoff modeling are suggested. 

68

7 Conclusions 

Four targets concerning the measurement of the two parameters ka and Ks were addressed in 

the present study: 

1. A portable air permeameter was developed to measure air permeability in situ, on-site 

(exhumed soil samples) and in the laboratory using two different sizes of core samples. 

The newly developed device performed well, and it was possible to carry out reliable 

measurements in all three situations. Results from measurements on both structured and 

unstructured soils showed that it was possible to make reliable in situ air permeability 

measurements using a newly developed shape factor model. 

2. The study showed that the choice of an appropriate representative elementary volume 

(REV) is important for studies of ka and kw. A significant difference in ka and kw meas- 

ured at two scales (100-cm 3 and 3140-cm 3 /6280-cm 3 soil samples) was found for struc- 

tured soils, showing higher values for the large soil samples. This scale-dependent differ- 

ence between sample size was less pronounced for the two sandy soils. In addition, ka and 

kw generally displayed higher variability for structured loamy soils compared to the un- 

structured sandy soils. Variability in both ka and kw in the sandy soils was significantly 

higher for the large (6280 cm 3 ) samples compared to the small (100 cm 3 ) samples for half 

of the individual horizons. No clear effect in the variability was observed between the two 

sample sizes for the structured loamy soils. For the structured soils the variability between 

measurements was lower for ka compared to the kw. The deviation between the two sam- 

ple sizes was most likely an effect of a non-representative sampling of larger macropores 

in the small 100-cm 3 cores. 

3. Linking water and air permeability functions has been one focus of soil physics research 

over the last half century. However, functional relationships as proposed e.g. by Brooks 

and Corey (1964) seem to be less feasible due to the different geometries and tortuosities 

of the gaseous and liquid phases. Using other more empirical methods of linking the two 

parameters might then be a better alternative. In general, a good relationship was found 

between Ks and ka measured at matric water potentials of −50 and −100 cm H2O respec- 

tively. The Ks-ka measurements in the present study supported those of Loll et al. (1999). 

The results in this study show that the relationship might be used on both large and small 

soil samples. However, a poor relationship for a well-sorted sandy soil illustrates the im- 

portance of the drainage of the soil samples when measuring ka on soils having a high 

frequency of medium-sized pores. 

69

4. The portable air permeameter proved to be an efficient tool for defining the structure of 

the spatial variability in an efficient and reliable way. This opens up for new methods of 

characterising the soil while obtaining new and valuable information about the soil vari- 

ability. ka,in situ measured in the topsoil in two small agricultural catchments (field slopes) 

showed that a spatial correlation of the log transformed values of ka existed, having a 

range of approximately 100 m. Additional measurement of ka,on-site (known boundary con- 

ditions) in an undisturbed constructed field in Japan indicated a spatial dependency of the 

log transformed data of approximately 20 m. 

Surface runoff modelling using a distributed GIS based model showed the importance 

of knowing the spatial variability of Ks estimated from measurements of ka. When in- 

creasing the resolution of Ks using a one layer Green and Ampt infiltration equation, a 

limit of 30-40 m was found for both field slopes. Below this limit the simulated runoff and 

hydrograph peaks were independent of resolution scale. A large effect on the output from 

the model was observed depending on whether the geometric average or the kriged values 

of ka were used. With only a few randomly chosen values of Ks to represent the variation 

in the catchment, a high scatter was seen in the output from the model compared to the 

situation where ka,in situ was measured in every grid point over the field slope. Since con- 

siderable measurement effort is necessary to get a satisfactory representation of the spatial 

variability of Ks at catchment scale, the use of in situ measurements of ka during periods 

when the soil-water content is close to field capacity is suggested as an alternative. 

70

8 Perspectives 

The portable air permeameter developed in this study performed well both when measuring in 

situ, on-site (exhumed soil samples) and in the laboratory using two different sizes of soil 

samples. Even though the instrument is portable, the limitation of the instrument is its rela- 

tively high weight because of the gas cylinder, which makes it difficult to carry the instrument 

over areas larger than field scale. A main improvement would be to develop an instrument 

where ka,in situ could be measured using less gas, which means a smaller gas cylinder could be 

used. A lower requirement for gas could be obtained by the use of more precise (but also 

more expensive) equipment for measuring gas flow and pressure difference. 

An increase in sample size from 100 cm 3 to 6280 cm 3 in the present study appears to 

improve the reliability of the ka or Ks measurements, especially for structured soils. Using 

soil samples of at least the same size as the REV will reduce the variability and improve the 

quality of the measurements. Therefore the REV with respect to ka for different soil types 

ought to be further examined. 

The measurement of Ks and ka in the present study supports the theory that a general 

Ks-ka relationship seems to exist when measuring on 100-cm 3 and 6280-cm 3 soil samples. 

The weakest relationship was found when measuring on the large soil samples, which proba- 

bly was related to the small number of replicates. A further examination of the relationship for 

large samples would then be suitable by measuring on a larger number of soil samples than 

this study. Measurements of ka in the laboratory on soil samples drained to a matric water 

potential of −50 cm H2O is relatively time-consuming due to a long drainage time. Therefore 

measurements of ka in the laboratory on soils sampled in the field at water content near field 

capacity could be a possible way of increasing the number of replicates. 

The results of the surface runoff simulation showed the importance of knowing the 

spatial variability of Ks estimated from measurements of ka. However, simulations were only 

carried out for one single constructed rainfall event. Further tests using different types of rain- 

fall events could increase the understanding of how the spatial variability of Ks affects the 

amount of surface runoff. However, a more important issue would be to evaluate simulation 

output from a measured runoff event in a catchment where both ka and Ks had been measured 

in a reliable way. 

71

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