X-ray attenuation techniques to explore the dynamics of water in porous media [Elektronische Ressource] / presented by Andreas Bayer

Sujets

• Physik

Informations

• Publié par: ruprecht-karls-universitat_heidelberg
• Publié le: 01 janvier 2005
• Nombre de lectures: 18
• Langue: Deutsch
• Poids de l'ouvrage: 5 Mo

Extrait

Dissertation
submitted to the
Combined Faculties for the Natural Sciences and for Mathematics
of the Ruperto Carola University of Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
presented by
Diplom-Physicist: Andreas Bayer
born in: Heidenheim a. d. Brenz
Oral examination: June 29, 2005X-ray attenuation techniques to explore the
dynamics of water in porous media
Referees: Prof. Dr. Kurt Roth
Prof. Dr. Bernd J ahneRontgenabschwachung zur Untersuchung der Dynamik von
Wasser in porosen Medien
Der Fluss von Wasser in einem ungesattigten porosen Medium wird von
der Richards’ Gleichung beschrieben. Diese gilt fur den Fall, dass die Luft
in den Poren beliebig mobil ist und jederzeit aus dem Porenraum entweichen
kann. Ausserdem benotigt man zur Losung der Richards’ Gleichung eine Pa-
rametrisierung fur die Beziehung zwischen Wassergehalt und Matrixpotenti-
al, die Boden-Wasser Charakteristik. Verschiedene Modelle beschreiben diese
Kurve, deren Parameter in der Regel durch inverse Modellierung dynamischer
Aus ussexp erimente bestimmt werden konnen.
Es wurde ein experimenteller Aufbau entwickelt, der es ermoglicht, Aus uss-
experimente an porosen Medien durchzufuhren, an denen gleichzeitig Ron tgen-
abschwachungspro le und tomographische Bilder gemessen werden konnen. Mit
diesem Aufbau wurden Experimente durchgefuhrt, mit denen die Gultigkeit
der Richards’ Gleichung bei verschiedenen Randbedingungen ub erpruft wer-
den konnte. Dazu wurden in der Probe gemessene vertikale Wasserverteilungen
mit Vorhersagen von Simulationen verglichen. Anhand der gefundenen Abwei-
chungen konnte gezeigt werden, dass die Kontinuitat der Luftphase nicht immer
gegeben ist, und Heterogenitaten die Objektivitat der gefunden e ektiv en Pa-
rameter verhindern konnen.
Ausserdem werden Methoden gezeigt, wie mit Hilfe von Ron tgentomographie
die grobe innere Struktur von Proben analysiert werden kann.
X-ray attenuation techniques to explore the dynamics of water
in porous media
The o w of water in an unsaturated porous medium is described by Richards’
equation. It is based on the assumption that the air in the pores is arbitrar-
ily mobile and able to disappear from the pore space at any time. To solve
Richards’ equation a parameterization of the relationship between the water
content and the matric potential is necessary: the soil-water characteristic.
There are several models that describe this curve. Their parameters are typi-
cally determined from dynamic out o w experiments.
A experimental setup was designed, that allows measurements of X-ray atten-
uation data and tomographic images during the out o w experiment. With this
setup experiments were done to test the validity of Richards’ equation for dif-
ferent boundary conditions. Therefore, the measured vertical distribution of
water was compared with simulated distributions. The di erences found showthat the continuity of the air phase is not given at all points, and heterogeneity
may prevent the determined e ectiv e parameters from being objective.
Additionally, methods are introduced to analyze the rough internal structure
of samples using X-ray computed tomography.Contents
1 Introduction 1
2 Theoretical background 5
2.1 Water o w through porous media . . . . . . . . . . . . . . . . . 5
2.1.1 Richards’ equation . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Methods to determine the pressure-saturation relation . 13
2.1.3 Monitoring pore structure and water content . . . . . . 14
2.2 X-ray production . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 X-ray attenuation in solid media . . . . . . . . . . . . . . . . . 18
2.3.1 Non-linear X-ray attenuation . . . . . . . . . . . . . . . 19
2.4 X-ray computed tomography . . . . . . . . . . . . . . . . . . . 22
2.4.1 Data pre-processing . . . . . . . . . . . . . . . . . . . . 23
2.4.2 Image reconstruction . . . . . . . . . . . . . . . . . . . . 25
2.4.3 The Fourier Slice Theorem . . . . . . . . . . . . . . . . 25
2.4.4 Filtered back-projection . . . . . . . . . . . . . . . . . . 26
2.4.5 Iterative reconstruction algorithm . . . . . . . . . . . . 30
2.4.6 Utilize the symmetry of the reconstruction problem . . 34
2.5 Vertical water content pro les . . . . . . . . . . . . . . . . . . . 37
2.6 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . 38
3 Experimental setup 41
3.1 X-ray system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.1 X-ray tube . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.2 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.3 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.4 Control software . . . . . . . . . . . . . . . . . . . . . . 45
3.1.5 Bow tie lter . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Multi-step out o w system . . . . . . . . . . . . . . . . . . . . . 48
iContents
4 Results and discussion 51
4.1 Measurement of the soil water retention curve with X-ray atten-
uation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.1 Sample preparation . . . . . . . . . . . . . . . . . . . . 52
4.1.2 Measurement protocol . . . . . . . . . . . . . . . . . . . 52
4.1.3 Estimation of the van Genuchten parameters from verti-
cal water content pro les . . . . . . . . . . . . . . . . . 54
4.1.4 Results of van Genuchten parameter estimation from X-
ray attenuation pro les . . . . . . . . . . . . . . . . . . 55
4.2 Water imbibition monitored with high temporal resolution . . . 57
4.2.1 Material, method and setup . . . . . . . . . . . . . . . . 57
4.2.2 Results and discussion . . . . . . . . . . . . . . . . . . . 58
4.3 Multi-step out o w and X-ray attenuation . . . . . . . . . . . . 61
4.3.1 Sample preparation . . . . . . . . . . . . . . . . . . . . 61
4.3.2 Measurement protocol . . . . . . . . . . . . . . . . . . . 61
4.3.3 Data processing . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.5 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4 Other applications of the X-ray system . . . . . . . . . . . . . . 73
4.4.1 Calibration of light transmission measurements with a
Hele-Shaw cell . . . . . . . . . . . . . . . . . . . . . . . 73
4.4.2 Structure analysis of natural soil columns . . . . . . . . 75
4.4.3 Water distribution in an arti cially structured medium
during in ltration . . . . . . . . . . . . . . . . . . . . . 78
5 Summary and conclusions 85
5.1 Restrictions and limits of the setup used for X-ray tomography 88
Bibliography 91
A Equations of the implemented reconstruction algorithm 99
Acknowledgments 103
iiList of Figures
2.1 Forces that act on a arbitrary volume of water . . . . . . . . . 6
2.2 Example curves for Brooks-Corey and van Genuchten parame-
terizations of h( ) and K( ) . . . . . . . . . . . . . . . . . . . 11
2.3 Ink-bottle e ect . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Examples of neutron tomography images . . . . . . . . . . . . 16
2.5 Non-linear attenuation measured for Cu and Al, and the result-
ing photon energy spectrum of the X-ray tube used . . . . . . 21
2.6 Fourier transformation of a parallel projection . . . . . . . . . . 27
2.7 Reconstructed images of a PVC phantom to demonstrate the
e ect of beam-hardening and its correction . . . . . . . . . . . 34
2.8 Utilize the symmetry of the reconstruction problem . . . . . . . 35
3.1 Dark current measurements for di eren t exposure times . . . . 43
3.2 Sketch of how to determine the positions of the relevant mechan-
ical parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Length of path through a circular shaped object and relative
intensity on detector . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Sketch of the bow tie lter used . . . . . . . . . . . . . . . . . . 47
3.5 The multi-step out o w setup . . . . . . . . . . . . . . . . . . . 48
4.1 Measured, inverse modeled and predicted out o w data . . . . 53
4.2 Vertical water content pro le within a sand sample at h = 3 cm 55
4.3 Experimental setup using a vertical detector orientation . . . . 57
4.4 Vertical water content pro les during imbibition in a dry sample 59
4.5 Simulated vertical water content pro les during imbibition in a
dry sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.6 Measured and simulated out o w data for a multi-step out o w
experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.7 Measured and simulated vertical water content pro les assuming
a homogeneous medium . . . . . . . . . . . . . . . . . . . . . . 66
iiiList of Figures
4.8 Vertical distribution of van Genuchten parameters assuming a
layered material model . . . . . . . . . . . . . . . . . . . . . . 67
4.9 Measured and simulated vertical water content pro les assuming
a layered material model . . . . . . . . . . . . . . . . . . . . . 68
4.10 Water retention curve and hydraulic conductivity for two sample
orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.11 Comparison of saturation data measured with X-ray attenuation
and light transmission . . . . . . . . . . . . . . . . . . . . . . . 74
4.12 Reconstructed slices through an undisturbed soil column at sev-
eral heights . .