Soil Physics
Lecture Notes
v2.2β, Autumn 2012
Kurt Roth
c 2012 by Kurt Roth
Copyright �
The right for non-commercial copies is herewith granted.
The current version of these notes is available from
www.iup.uni-heidelberg.de/institut/forschung/groups/ts/students/sp
Please
Do not put a copy of these notes on your homepage
but rather link to the original site
and cite them as
Roth, K., 2012: Soil Physics. Lecture Notes.
Institute of Environmental Physics, Heidelberg University
giving the above link whenever possible.
Contact
Kurt Roth
Institute of Environmental Physics, INF 229
Heidelberg University
D-69120 Heidelberg
kurt.roth@iup.uni-heidelberg.de
Cover
Irrigated agricultural field in the North China Plain near Fengqiu, Henan
Province, China.
Preface
Soil physics aims at understanding physical processes in terrestrial environments and at the quantitative prediction of the associated phenomena.
This is challenging because the underlying processes are often nonlinear and
coupled, and they run in hierarchical heterogeneous media with hardly known
architectures. These media – porous structures with intricate geometries
that originate from various generators operating on different length scales –
set terrestrial systems apart from other large environmental compartments
like the atmosphere or the ocean. In these other compartments, multiscale phenomena almost exclusively arise from the dynamics of the process,
typically from fluid dynamics, whereas the boundary conditions are irrelevant
to a great extent.
While the physics of most processes relevant for soils is understood reasonably well at the scale of individual pores, this is no more the case at the larger
scales of a soil profile. As scales increase even further, eventually to entire
landscapes, ever more heuristic approximations have to be introduced in order
to arrive at palpable results. These in turn are mandatory for understanding
and handling a great many environmental and engineering problems.
These notes attempt to explore the transition from fundamental understanding to heuristic description and quantitative approaches to larger scales.
The thematic focus is on transport processes, in particular on subsurface
flow of water, on transport of dissolved substances, plant nutrients as well as
groundwater contaminants, and on the movement of thermal energy. These
processes are studied at scales ranging from a few hundred micrometers
to a few hundreds of meters. Much smaller and much larger scales are
only considered occasionally. Notable omissions in the spectrum of physical processes include soil mechanics, soil erosion, surface water, and soilvegetation-atmosphere interactions. Also, experimental methods are not
represented to the depth that would do justice to their importance in a
still essentially explorative discipline. I should acknowledge and emphasize
that transport processes, while being an essential and in some sense primary
aspect of soil systems, are but one class of processes relevant for the function
of soils in the environment. Much larger classes are formed by chemical
interactions, microbiological processes, and by the actions of macroflora and
-fauna.
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ii
Preface
The notes are written from a rather fundamental perspective but attempt
to keep applications in mind. They try to strike a balance between clean
mathematical formulations and simple intuitive concepts, with necessarily
many compromises on both ends. The material is aimed at advanced undergraduate and graduate students in environmental sciences, physics, and
engineering and it does require some passion for mathematics. Some problems
are added at the end of each chapter. They fall into two classes: (i) simple
calculations, just applying some equations given in the text, or marginal
extensions of the theory and (ii) qualitative explorations of complicated
phenomena which demand bold assumptions and may only yield order of
magnitude answers. The first class is aimed at exercising the mechanics of the
theory while the second one is to entice the reader to plough deeper.
Although the roots of these notes reach back to 1993, they continue to
evolve, with the material still not adequately balanced and certainly containing a number of suboptimal explanations, mistakes, and errors, hopefully not
too many. I am most grateful for any and all suggestions on how to correct
and improve things such that they best serve the reader.
How to Work through these Notes
The chapters following the introduction are arranged such that fundamental processes are studied first, bringing
there representation from the pore scale to the continuum scale, and only
then are they explored as they work in soils. However, the chapters are
rather independent of each other, with ample cross-references where previous
results are required, such that they may be worked through along different
lines. With the priority on the continuous development of the thematic fields,
the sequence 2, 3, [5], 6, 4, 7, 8 may be optimal, whereas with a priority on
increasing difficulty, the sequence 3.1, 3.3, 4, 5, 7, [8], 2, 3.2, 6 is more
appropriate.
Acknowledgements
I thank a number of colleagues for providing critical
comments, data sets, images, and numerical solvers: Hans-Jörg Vogel, now
at Umweltforschungszentrum (UFZ), Halle, for the images on soil crosssections, reconstructed three-dimensional representations, and innumerable
discussions on soil hydraulics; Ute Wollschläger, now at UFZ, Leipzig, for
many of her research results on groundwater flow in the Rhein-Neckar region;
Dietmar Wagenbach, Institute of Environmental Physics (IUP), Heidelberg
University, for critically reading the chapter on fluid dynamics; Ed Sudicky,
University of Waterloo, for the Borden data set; Isabelle Cousin, Institut
National de la Recherche Agronomique, Orléans, for the three-dimensional
dataset from microscopic cross-sections of a soil sample. Whenever satellite
images were needed for illustration, modis.gsfc.nasa.gov was my source.
The numerical simulations of groundwater flow were run with an AMG
(algebraic multi-grid) solver developed by Peter Bastian and Olaf Ippisch,
Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University. Those of flow in the vadose zone were done with SWMS, the precursor of
HYDRUS, by Jirka Šimu̇nek, UC Riverside.
iii
Preface
Finally, I am glad to acknowledge my students for everything from pointing
out errors to suggesting improvements. Most importantly though, I appreciate their interest, critical questions, and inspiring after-class discussions
which makes teaching so rewarding.
Heidelberg, October 2012
Kurt Roth
Contents
1 Introduction
1.1 Global Terrestrial Water and Energy Fluxes
1.2 Anthropogenic Use of Water . . . . . . . . .
1.2.1 Water Quantity . . . . . . . . . . . .
1.2.2 Water Quality . . . . . . . . . . . . .
1.3 Why Study Small-Scale Processes in Soils? .
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2 Fluid Dynamics
2.1 Generic Formulation of Single Phase Dynamics .
2.1.1 Forces and Fluxes . . . . . . . . . . . . . .
2.1.2 Conservation Laws . . . . . . . . . . . . . .
2.1.3 Dynamics . . . . . . . . . . . . . . . . . . .
2.2 Material Properties . . . . . . . . . . . . . . . . .
2.2.1 Density . . . . . . . . . . . . . . . . . . . .
2.2.2 Viscosity . . . . . . . . . . . . . . . . . . .
2.3 Flow of Incompressible Newtonian Fluid . . . . .
2.3.1 Dynamics . . . . . . . . . . . . . . . . . . .
2.3.2 Approximations of Navier-Stokes Equation
2.3.3 Similarity Analysis . . . . . . . . . . . . . .
2.3.4 Initial and Boundary Conditions . . . . . .
2.3.5 Particular Solutions . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
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3 Fluids in Porous Media
3.1 Architecture of Porous Media . . . . . . .
3.1.1 Simple Porous Media . . . . . . . .
3.1.2 Sediments . . . . . . . . . . . . . . .
3.1.3 Soils . . . . . . . . . . . . . . . . . .
3.2 Multiple Phases . . . . . . . . . . . . . . .
3.2.1 Interfacial Energy and Tension . . .
3.2.2 Discontinuity of Pressure . . . . . .
3.2.3 Equilibrium Distribution of Multiple
3.3 Transition to Continuum Scale . . . . . . .
3.3.1 Representative Elementary Volume .
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vi
Contents
3.3.2 Texture . . . . . . .
3.3.3 State Variables . . .
3.3.4 Mass Balance . . . .
3.3.5 Empirical Flux Law
3.4 Material Properties . . . .
3.4.1 Capacity . . . . . .
3.4.2 Conductivity . . . .
3.4.3 Flux Law . . . . . .
3.4.4 Compressibility . . .
Exercises . . . . . . . . . .
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4 Solutes in Porous Media
4.1 Transport at the Pore-Scale . .
4.1.1 Molecular Diffusion . . .
4.1.2 Taylor-Aris Dispersion . .
4.1.3 Dispersion in Pore-Space
4.2 Transport in Porous Media . . .
4.2.1 State Variables . . . . . .
4.2.2 Mass Balance . . . . . . .
4.2.3 Empirical Flux Law . . .
4.2.4 Dynamics . . . . . . . . .
4.2.5 Material Properties . . .
Exercises . . . . . . . . . . . . .
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5 Groundwater Flow
5.1 Dynamics of Flow in Confined Aquifer . .
5.1.1 Stationary Flow . . . . . . . . . . .
5.1.2 Large Aquifers . . . . . . . . . . . .
5.2 Stationary Flow in Uniform Aquifer . . . .
5.2.1 Pumping Well in Regional Flow . .
5.2.2 Dipole Pumping in Regional Flow .
5.3 Stationary Flow in Heterogeneous Aquifer
5.3.1 A Field Study: The Borden Site . .
5.4 Simulated Single-Scale Media . . . . . . .
5.4.1 Hydraulic Structure . . . . . . . . .
5.4.2 Simulation of Stationary Flow . . .
5.4.3 Effective Hydraulic Conductivity . .
5.5 Dynamics of Flow in Unconfined Aquifer .
5.6 Case Study: Groundwater in Rhein-Neckar
5.6.1 Groundwater Model . . . . . . . . .
5.6.2 Model Calibration . . . . . . . . . .
5.6.3 Simulation . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . .
vii
Contents
6 Soil Water Flow
6.1 Dynamics of Soil Water . . . . . . . . . . .
6.1.1 Degenerate Multiphase Regime . . .
6.1.2 Continuous Multiphase Regime . . .
6.1.3 Discontinuous Multiphase Regime .
6.2 Stationary Flow . . . . . . . . . . . . . . .
6.2.1 Uniform Soil . . . . . . . . . . . . .
6.2.2 Layered Soil . . . . . . . . . . . . .
6.2.3 Heterogeneous Soil . . . . . . . . . .
6.2.4 Effective Dynamics . . . . . . . . . .
6.3 Transient Flow . . . . . . . . . . . . . . .
6.3.1 Infiltration . . . . . . . . . . . . . .
6.3.2 Drainage . . . . . . . . . . . . . . .
6.3.3 Infiltration Event . . . . . . . . . . .
6.3.4 Evapotranspiration . . . . . . . . . .
6.3.5 Natural Atmospheric Forcing . . . .
6.4 Inverse Estimation of Hydraulic Properties
6.5 Preferential Flow . . . . . . . . . . . . . .
6.5.1 Macropore Flow . . . . . . . . . . .
6.5.2 Flow Instabilities . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . .
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7 Solute Transport
7.1 Transport with Stationary Groundwater Flow
7.1.1 Generic Relations . . . . . . . . . . . .
7.1.2 Near-Field: Stochastic Convection (SC)
7.1.3 Far-Field: Convection-Dispersion (CD)
7.1.4 Transfer Functions . . . . . . . . . . . .
7.1.5 A First Glimpse at Heterogeneity . . . .
7.1.6 Heterogeneous Media . . . . . . . . . .
7.1.7 Aquifers . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . .
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8 Soil Heat
8.1 Thermodynamics of Water . . . . . . . . . . . . . . . .
8.1.1 Free Water . . . . . . . . . . . . . . . . . . . . .
8.1.2 Water in Porous Medium . . . . . . . . . . . . .
8.2 Heat Conduction in Solids . . . . . . . . . . . . . . . .
8.2.1 Dynamics of Conductive Heat Transfer . . . . .
8.2.2 Ground Temperature as Archive for Paleoclimate
8.3 Heat Conduction in Porous Media . . . . . . . . . . . .
8.3.1 Effective Material Properties . . . . . . . . . . .
8.3.2 Effective Dynamics . . . . . . . . . . . . . . . . .
8.4 Permafrost Soil . . . . . . . . . . . . . . . . . . . . . .
8.4.1 High-Latitude Permafrost . . . . . . . . . . . . .
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viii
Contents
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
ix
Contents
A Mathematical Tools
A.1 Vector Analysis . . . . . . . . . . . . . . . . . . . . .
A.1.1 Gauss Theorem . . . . . . . . . . . . . . . . . .
A.1.2 Total Derivative . . . . . . . . . . . . . . . . .
A.1.3 Conservation Laws . . . . . . . . . . . . . . . .
A.1.4 Stream Function . . . . . . . . . . . . . . . . .
A.1.5 Lagrangian Multipliers . . . . . . . . . . . . . .
A.2 Statistical Descriptions . . . . . . . . . . . . . . . . .
A.2.1 Random Variables . . . . . . . . . . . . . . . .
A.2.2 Central Limit Theorem . . . . . . . . . . . . .
A.2.3 Random Functions . . . . . . . . . . . . . . . .
A.2.4 Geostatistics . . . . . . . . . . . . . . . . . . .
A.3 Laplace Transform . . . . . . . . . . . . . . . . . . .
A.3.1 Transforms of Derivatives and Integrals . . . .
A.3.2 Transformed Solution of Differential Equations
A.3.3 Statistical Moments . . . . . . . . . . . . . . .
A.3.4 Inverse Laplace Transformation . . . . . . . . .
A.3.5 Table of Laplace Transform Pairs . . . . . . . .
A.4 Fourier Transform . . . . . . . . . . . . . . . . . . . .
A.5 Differential Equations . . . . . . . . . . . . . . . . . .
A.5.1 Ordinary Differential Equations . . . . . . . . .
A.5.2 Partial Differential Equations . . . . . . . . . .
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291
291
291
291
292
294
294
295
295
296
297
305
312
313
314
315
316
319
321
322
323
323
B Numerical Methods
B.1 Partial Differential Equations .
B.1.1 Finite Differences . . . . .
B.1.2 Finite Elements . . . . .
B.2 Nonlinear Parameter Estimation
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325
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329
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C Modeling Exercises
C.1 Stationary Water Flow in Layered Soil . . . . .
C.2 Simple Atmospheric Forcing of Soil Water Flow
C.3 Fluctuating Water Table . . . . . . . . . . . . .
C.4 Solute Transport with Stationary Water Flow .
C.5 Flow and Transport under Irrigated Field . . .
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333
335
338
340
341
344
D Solutions to Problems
D.1 Problems in Chapter
D.2 Problems in Chapter
D.3 Problems in Chapter
D.4 Problems in Chapter
D.5 Problems in Chapter
D.6 Problems in Chapter
D.7 Problems in Chapter
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349
349
351
357
358
358
360
362
2
3
4
5
6
7
8
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x
Contents
E Some Constants and Material Properties
E.1 General Constants . . . . . . . . . . . . . .
E.2 Material Properties of Water . . . . . . . .
E.3 Material Properties of Air . . . . . . . . .
E.4 Properties of Geologic Materials . . . . . .
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365
365
365
366
366
Dictionary English-Deutsch and Glossary
367
Bibliography
371
Index
381
List of Figures
1.1
1.2
1.3
1.4
1.5
Schematic of global terrestrial water cycles . . . . . . . . .
Schematic of global terrestrial energy fluxes . . . . . . . .
Huang He (Yellow river) at Lanzhou . . . . . . . . . . . .
Aral region as seen from the MODIS sensor aboard Terra
Arsenic concentrations in the groundwater of Bangladesh
2.1
2.2
2.3
2.4
2.5
2.6
Fluid element . . . . . . . . . . . . . . . . . . . . . . . . . . .
Momentum flux in a fluid . . . . . . . . . . . . . . . . . . . .
Classical Gedankenexperiment on viscosity . . . . . . . . . . .
Relation between shear stress and strain rate for different fluids
Forces on fluid element with Stokes flow . . . . . . . . . . . .
Velocity field for laminar flow in long circular cylinder . . . .
14
15
18
20
26
29
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
Artificial porous medium created from sieved sand . . . . . .
Braided river as precursor of a sedimentary reservoir . . . . .
Multi-scale architecture of a typical sedimentary structure . .
Fractured rock . . . . . . . . . . . . . . . . . . . . . . . . . .
Micropore system of a loamy-clay soil . . . . . . . . . . . . .
Macropore system of a loamy-clay soil . . . . . . . . . . . . .
Three-dimensional reconstruction of pore space . . . . . . . .
Schematic for interfacial forces . . . . . . . . . . . . . . . . .
Interfacial tension and energy . . . . . . . . . . . . . . . . . .
Discontinuity of pressure across interface . . . . . . . . . . . .
Interfacial tensions and contact angle in a system with a solid,
a liquid, and its vapor . . . . . . . . . . . . . . . . . . . . . .
Interfacial tensions in a water-oil-air system . . . . . . . . . .
Rise of liquid in smooth-walled capillary . . . . . . . . . . . .
Equilibrium points for liquid in corrugated capillary . . . . .
Transition from pore-scale to continuum representation . . . .
REV for the porosity of the pore space shown in Figure 3.6 .
Flow through macroscopic planar REV . . . . . . . . . . . . .
Macroscopic flux and driving force in anisotropic media . . .
Hydraulic capacity of porous medium illustrated for a bundle
of capillaries . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
33
34
35
36
37
37
39
40
41
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
xi
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3
4
6
7
8
42
43
44
45
47
48
54
55
58
xii
List of Figures
3.20 Hysteresis of hydraulic capacity . . . . . . . . . . . . . . . . .
3.21 Experimental setup used by Topp and Miller [1966] for the
direct measurement of soil hydraulic properties . . . . . . . .
3.22 Pressure-saturation relation . . . . . . . . . . . . . . . . . . .
3.23 Brooks-Corey, van Genuchten, and modified van Genuchten
parameterizations of Θ(hm ) . . . . . . . . . . . . . . . . . . .
3.24 Pressure-conductivity relation . . . . . . . . . . . . . . . . . .
3.25 Saturation-conductivity relation . . . . . . . . . . . . . . . . .
3.26 Soil hydraulic properties for various soil textures with Mualemvan Genuchten and Mualem-Brooks-Corey parameterization .
3.27 Scaled hydraulic conductivity functions . . . . . . . . . . . .
3.28 Sketch of weakly compressible media . . . . . . . . . . . . . .
3.29 Two grains of sand with water between them . . . . . . . . .
58
4.1
4.2
78
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.1
5.2
5.3
5.4
5.5
Two-dimensional Brownian motion . . . . . . . . . . . . . . .
Convection dominated transport of Brownian particles in a
cylindrical tube with laminar flow . . . . . . . . . . . . . . . .
Probability distribution function for particles undergoing TaylorAris dispersion . . . . . . . . . . . . . . . . . . . . . . . . . .
Temporal evolution of apparent parameters in Taylor-Aris dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mixing in junction of pore network . . . . . . . . . . . . . . .
Parallel stream-tubes as an approximation to transport over
short distances . . . . . . . . . . . . . . . . . . . . . . . . . .
PLIF-measured molecular diffusion in porous medium . . . .
PLIF-measured transport of dye tracer with constant flow . .
Experimentally determined relation between longitudinal dispersion coefficient and Peclet number . . . . . . . . . . . . . .
59
60
63
65
66
68
70
73
75
82
83
86
87
93
97
98
99
Sketch of a multi-story valley aquifer . . . . . . . . . . . . . . 102
Sketch of two-story groundwater system . . . . . . . . . . . . 102
High Plains aquifer . . . . . . . . . . . . . . . . . . . . . . . . 105
Sketch for calculating the pressure field of a single pumping well106
Isobars and stream function for a single pumping well in a
uniform regional flow field . . . . . . . . . . . . . . . . . . . . 107
5.6 Isobars and stream function for dipole pumping in a uniform
regional flow field . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.7 Histograms of permeability at Borden site . . . . . . . . . . . 110
5.8 Spatial distribution of permeability at Borden site . . . . . . 111
5.9 Simulated two-dimensional flow at the Borden site . . . . . . 112
5.10 Probability distribution functions of log10 (K) used for the
simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.11 Small section from the log-conductivity fields . . . . . . . . . 114
5.12 Flow in macroscopically uniform media with different degrees
of small scale heterogeneity . . . . . . . . . . . . . . . . . . . 116
List of Figures
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
5.24
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
High- and low-flux regions in macroscopically uniform media 117
Range of water content in vadose zone and unconfined aquifer 119
Definition sketch for flow in unconfined aquifer . . . . . . . . 120
Schematic of architecture and water flow in Rhein-Neckar region122
Absolute positions of main hydrogeologic units in Rhein-Neckar
region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Hydrogeologic cross-section through Rhein valley . . . . . . . 124
Thickness of aquitard 1 and distribution of facies . . . . . . . 124
Estimated mean annual groundwater recharge in Rhein-Neckar
region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Measured hydraulic head in aquifers 1 and 2 . . . . . . . . . . 127
Representations of aquifer 1 and 2 in numerical groundwater
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Calibration results of groundwater model for Rhein-Neckar
region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Simulation of stationary groundwater flow in Rhein-Neckar
region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Flow regimes in soils . . . . . . . . . . . . . . . . . . . . . . . 134
Dimensionless matric head for stationary flow in uniform medium
with constant water table . . . . . . . . . . . . . . . . . . . . 141
Salt precipitation in arid mountain valleys near Qinghai Hu,
China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Maximum evaporation flux sustainable as function of layer
thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Typical two-layer soil . . . . . . . . . . . . . . . . . . . . . . . 145
Soil hydraulic properties used for Figures 6.7–6.12 . . . . . . 146
Matric head, water content, and hydraulic conductivity for
stationary infiltration and evaporation in two-layer profile silt
on top of sand . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Sketch for shape of hm (z) near an interface . . . . . . . . . . 148
Same as Figure 6.7 but for sand on top of silt . . . . . . . . . 149
Schematic action of an inclined capillary barrier . . . . . . . . 150
Hydraulic state variables for stationary low-flux infiltration
into heterogeneous soil with irregular layers . . . . . . . . . . 152
Water saturation for stationary high-flux infiltration into soil
with heterogeneous layers . . . . . . . . . . . . . . . . . . . . 154
Matric head and water saturation for stationary evaporation
from soil with heterogeneous layers . . . . . . . . . . . . . . . 155
Patterns of precipitated salt . . . . . . . . . . . . . . . . . . . 156
Average and effective quantities for a one-dimensional representation of the heterogeneous layers . . . . . . . . . . . . . . 158
Same as Figure 6.15 but for evaporative flow . . . . . . . . . 160
Soil water velocity, diffusivity, and dispersivity for Mualem-van
Genuchten parameterization . . . . . . . . . . . . . . . . . . . 161
xiii
xiv
List of Figures
6.18
6.19
6.20
6.21
6.22
6.23
6.24
6.25
6.26
6.27
6.28
6.29
6.30
6.31
6.32
6.33
6.34
6.35
6.36
6.37
7.1
7.2
7.3
7.4
7.5
Sketch for shape of hm (z) near an interface . . . . . . . . . .
Transient infiltration into uniform sand and silt . . . . . . . .
Transient infiltration into two-layer soil . . . . . . . . . . . .
Drainage of sand and silt after end of constant infiltration . .
Propagation of single infiltration pulse into uniform dry soil .
Evaporation and transpiration from uniform sand and silt . .
Root distribution density and root activity . . . . . . . . . .
Water content in sand and silt for realistic atmospheric forcing
Profiles of θ and hm in sand and silt for the most intense rain
event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Statistical distribution of θ during the entire period shown in
Figure 6.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Typical setup for multi-step outflow experiment . . . . . . . .
Typical result of a multi-step outflow experiment . . . . . . .
Mualem-van Genuchten parameterization for parameters given
in Table 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conceptual setup for evaporation experiment . . . . . . . . .
Flow instability induced at transition from fine- to coarsetextured sand observed in Hele-Shaw cell . . . . . . . . . . . .
Evolution of hydraulic states during passage of finger tip . . .
Dynamic capillary pressure . . . . . . . . . . . . . . . . . . .
Pressure drop across porous column as interface between two
immiscible fluids passes through . . . . . . . . . . . . . . . . .
Saturation overshoot for different fluxes . . . . . . . . . . . .
Fingered flow in medium with heterogeneous layer at some
depth interval . . . . . . . . . . . . . . . . . . . . . . . . . . .
162
163
165
166
168
170
171
173
173
174
177
178
179
181
186
187
188
190
190
191
Relation between total concentration and solute flux . . . . . 198
Dimensionless travel distance pdf for CD process . . . . . . . 204
Dimensionless travel time pdf for CD process . . . . . . . . . 206
Evolution travel distance pdfs with SC- and CD-process . . . 207
Projection of solute flux to downstream location using transfer
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.6 Space-time diagram for interaction between mobile and immobile region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
7.7 Travel time pdf and cdf for MIM model. . . . . . . . . . . . . 215
7.8 Travel distance pdf for MIM model with different rate parameters217
7.9 Travel distance pdf for MIM model for Ω = 0.25 after short
and long travel times . . . . . . . . . . . . . . . . . . . . . . . 218
7.10 Sketch to explain immobile water in saturated, uniform, granular media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
7.11 Initial phase of solute transport shown in Figure 7.12 . . . . . 222
7.12 Evolution of concentration distributions from Figure 7.11 for
longer times . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
List of Figures
xv
7.13 Transition pdf in medium with var(log10 (K/K0 ) = 0.15 and
dispersion by molecular diffusion and hydromechanic dispersion with λ� = 0.005 m and 0.025 m . . . . . . . . . . . . . . 225
7.14 Horizontal component of particle distribution’s center of gravity and apparent velocities . . . . . . . . . . . . . . . . . . . . 228
7.15 Apparent global dispersivities for the simulations shown in
Figures 7.12–7.13 . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.16 Same as Figure 7.12 but for var(log10 (K/K0 )) = 0.75 . . . . . 232
7.17 Apparent global dispersivities for the simulations with var(log10 (K/K0 )) =
0.75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
7.18 Same as Figure 7.12 but for var(log10 (K/K0 )) = 3.77 . . . . . 234
7.19 Evolution of point source from Figure 7.18 but with λ� = 0.005 m235
7.20 Apparent longitudinal dispersivity increasing with scale of experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
8.11
8.12
8.13
8.14
8.15
8.16
8.17
8.18
8.19
8.20
8.21
8.22
8.23
Schematic phase diagram of free water . . . . . . . . . . . . . 244
Chemical potential of solid, liquid, and vapor phase of water . 247
Experimentally determined soil freezing characteristic . . . . 248
Transfer function pt (t; z) for pure diffusion . . . . . . . . . . . 252
Temperatures measured in a deep borehole . . . . . . . . . . 255
Reconstruction of the mean global temperature for the Northern hemisphere . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Projection of surface temperature to greater depths . . . . . . 258
Temperature anomaly for forcing by Mann-Jones reconstruction259
Mathematica code for solving heat conduction problem with
spatially varying thermal diffusivity . . . . . . . . . . . . . . . 260
Temperature anomaly in non-uniform medium for forcing by
Mann-Jones reconstruction . . . . . . . . . . . . . . . . . . . 261
Sketch for dependence of thermal conductivity on water content263
Liquid water content and effective thermal capacity for idealized soil freezing characteristic . . . . . . . . . . . . . . . . . 264
Vapor diffusion in partly saturated porous medium . . . . . . 267
Enhancement factor in parameterization of effective thermal
conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
Transfer function pt (t; z) for pure diffusion . . . . . . . . . . . 273
Notations for permafrost soils . . . . . . . . . . . . . . . . . . 275
Distribution of permafrost on the Northern hemisphere . . . . 276
Mud boils at the Bayelva permafrost site . . . . . . . . . . . . 278
Thermal and hydraulic dynamics monitored at Bayelva site,
Svalbard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Typical thermal periods at a high-latitude permafrost site . . 280
Projection of surface temperatures to bottom of active layer . 283
Effective conductive heat flux at Bayelva site . . . . . . . . . 285
Production of latent heat estimated from measured temperatures286
xvi
List of Figures
A.1
A.2
A.3
A.4
A.5
A.6
A.7
A.8
A.9
Conservation from Lagrangian perspective . . . . . . . . . . .
Constrained optimization using Lagrangian multipliers . . . .
Four perspectives on a random function . . . . . . . . . . . .
Ensemble averages of random function . . . . . . . . . . . . .
ergodic random function . . . . . . . . . . . . . . . . . . . . .
Sketch of a typical semivariogram . . . . . . . . . . . . . . . .
Estimation variance for interpolation between three locations
Geostatistical interpolation based on three data points . . . .
Geostatistical simulation based on three data points . . . . .
293
295
298
299
304
306
308
310
311
B.1 Discretization of domain for finite difference method . . . . . 326
B.2 Basis functions for the finite element method . . . . . . . . . 328
B.3 Sketch for the distortion of parameter space {pi } . . . . . . . 331
C.1 Mathematica code for calculating matric head for stationary
flow in layered medium . . . . . . . . . . . . . . . . . . . . . . 336
C.2 Matric head hm (z) for various stationary water fluxes in layered medium . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
D.1
D.2
D.3
D.4
Evolution of function cos(x) under propagation operator −u∂x u350
Notation for the solution of Exercise 3.2 . . . . . . . . . . . . 352
Dynamics of capillary rise . . . . . . . . . . . . . . . . . . . . 354
Sketch for the solution of Exercise 3.8 . . . . . . . . . . . . . 354
List of Tables
1.1 Consumption of water for the production of food and other goods
3.1 Mualem-van Genuchten parameters for hydraulic properties . .
5
68
5.1 Some parameters of the hydraulic conductivity distributions . 114
5.2 Some characteristics of the flow fields shown in Figure 5.12 . . 115
5.3 Range for hydraulic parameters of topmost hydrogeologic units
in Rhein-Neckar region . . . . . . . . . . . . . . . . . . . . . . 125
6.1 Mualem-van Genuchten parameters for hydraulic properties used
for Figures 6.7–6.12 . . . . . . . . . . . . . . . . . . . . . . . . 146
6.2 Values for hydraulic state variables in two-layer soil at static
equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.3 Optimal values of Mualem-van Genuchten parameters from Figure 6.29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.1 Typical values for mass density, specific heat capacity, and thermal conductivity of some constituents of geologic formations . 250
8.2 Typical damping depth for fluctuations with different periods . 256
xvii
Symbols
This list contains the most important symbols and notations used. It is still
rather incomplete and will be upgraded in a future version. Where possible,
a reference (equation and page number) is given and the unit is indicated in
brackets.
The mathematical structure of symbols is indicated by their typographical
appearance:
a scalar
� vector, unit length vector
a, a
A tensor
sin standard function
Subscripts usually refer to a component of a vector (x, y, z, or 1, 2, 3) or to
the phase (g, w, s). When dealing with quantities that refer to different scales,
the superscripts µ and m are used to indicate the micro- and the macroscale,
respectively. They are suppressed when only one scale is considered.
Parentheses (. . . ) are used for arguments of functions and operators, brackets [. . . ] group terms or indicate concentrations of a chemical component, and
curly brackets {. . . } collect operators.
The arguments of functions are suppressed if they are clear from the
context, e.g., ∂x f instead of ∂x f (x). They are written, however, if the dependence on an argument is emphasized, e.g., Kd (Cw ) for a nonlinear adsorption
isotherm. Similarly, derivatives are written as shorthand operators: ∂x , ∂xx
for the first and second partial derivative with respect to x, respectively, and
dt for the total derivative ∂t + v · ∇ with respect to time. If the derivative is
∂�
to be emphasized, the long form ∂x
is used.
Sign Convention
The normal vector n on the surface of some volume points outwards. The
vertical (z) axis points downward, in the direction of the acceleration of
gravity. Its origin is typically chosen at the soil surface. Accordingly, z is
called the depth.
Lowercase Latin Symbols
g
acceleration of gravity [m s−2 ]
hm matric head [m]
j
flux; quantity X flowing through unit area per unit time
[X m−2 s−1 ]
js
mass flux of solute [mol m−2 s−1 ]
jw
volume flux of water [m s−1 ]
�
n
unit normal vector
microscopic Peclet number
q
flow [X s
r
position [m]
v
velocity [m s−1 ]
x
position [m]
pe
−1
3.41
62
3.24
52
4.15
81
]
Uppercase Latin Symbols
A
hydraulic gradient
Ca
capillary number
5.9 105
6.48 189
3
Cm concentration in mobile phase [mol m ]
7.40 210
3
Cim concentration in immobile phase [mol m ]
7.40 210
3
Ct
total concentration [mol m ]
Cs
concentration in sorbed phase [mol kg3 ]
Cw concentration in water phase [mol m3 ]
D
dispersion tensor [m2 s−1 ]
F
force [kg m s−2 ]
Fr
Froude number
6.4 135
−1
5 −1
2.24
22
3.32
55
54
K
hydraulic conductivity tensor; [J
form or [m s−1 ] in head form
k
permeability tensor [m2 ]
3.30
R
retardation factor
7.43 211
Re
Reynolds number
2.24
S
m s
] in potential
22
linear system
7.34 207
Pe
Peclet number
7.25 203
St
Strouhal number
2.24
22
V
generalized velocity operator
4.42
90
Lowercase Greek Symbols
ψw
water potential, potential energy of water [J m−3 ]
ψm matric potential [J m
−3
]
6.1 135
3.19
51
ρb
bulk density of porous medium [kg m−3 ]
3.13
49
ρm
mass density of solid matrix [kg m−3 ]
3.14
49
3.21
52
ρw
mass density of water [kg m
θ
θm
volume fraction [–]
mobile water content [–]
−3
]
3.11 46
7.40 210
θim immobile water content [–]
7.40 210
θw
3.23
52
3.15
49
volumetric water content [–]
Uppercase Greek Symbols
Θ
saturation [–]
Mathematical Notation
erf
error function
erfc complementary error function
var variance
∂t
partial derivative with respect to time [s−1 ]
∂x
partial derivative with respect to spatial coordinate x
[m−1 ]
second partial derivative with respect to spatial
coordinate x [m−2 ]
total derivative with respect to time [s−1 ]
∂xx
dt
∇
partial derivative with respect to space [m−1 ]
L
Laplace transform
∇h horizontal partial derivative with respect to space [m−1 ]
f�
Laplace transform of function f
L
�
−1
inverse Laplace transform
a generic argument