environment west of Berlin (Germany) and coarse sand samples with a ..... This
MSc thesis has been made at the department of applied geophysics at the ...
SATURATED HYDRAULIC CONDUCTIVITY DERIVED FROM (HYDRO)GEOPHYSICAL MEASUREMENTS
Stefan Kooman, Berlin, May 2003
SATURATED HYDRAULIC CONDUCTIVITY DERIVED FROM (HYDRO)GEOPHYSICAL MEASUREMENTS
‘Flow through the porous medium “ground coffee” is one of the most important when catching a deadline’ S. Kooman, Berlin 2003
Stefan Kooman, Reg. Nr.: 780519463100 Course Code: K150-707, 30 ECTS
Supervisors: Henny van Lanen, Sub-Department of Water Resources, WUR Martin Müller, Applied Geophysics, TU-Berlin
Applied Geophysics
ii
Abstract Surface nuclear magnetic resonance (SNMR) is a geophysical technique that has proven to be a useful tool for the investigation of hydrologic properties of aquifers (porosity φ , saturated hydraulic conductivity, k) in the past few years. To enable an enhanced understanding and interpretation of SNMR data, laboratory nuclear magnetic resonance (NMR) properties of synthetic and natural unconsolidated samples were analyzed (artificial glass pearls, sand mixtures, bore-core samples from a Quaternary environment west of Berlin (Germany) and coarse sand samples with a varying clay content). Until recently laboratory NMR only focused on consolidated sediments, in this research an attempt is made to investigate unconsolidated sediments. To verify the NMR measurements, pore space properties have been analyzed (specific surface, porosity). Frequency dependent electrical conductivity (FDEC) measurements were carried out to investigate the relation with NMR. Finally hydraulic conductivity measurements were conducted to verify the hydraulic conductivity derived from NMR relaxation time and hydraulic conductivity estimates based on grain size (after Hazen, Kozeny-Carman). The results show that the relation used to obtain hydraulic conductivity from NMR relaxation time is suitable to predict the saturated hydraulic conductivity for clay and coarse sand samples. Furthermore it is found that the predictions of hydraulic conductivity for intermediate grain sizes with different clay content over a wide range vary. A more individual approach with regard to the paramagnetic properties of the material might be needed to allow successful estimations. Clay (because of its high specific surface, approx 35 m 2 /gr, and high surface-to-pore-volume, S/V, ≈ 60 µ m −1 ) has a big influence on the NMR relaxation time and the associated saturated hydraulic conductivity. The NMR relaxation times of a series of coarse sand samples (d 1.0-0.5 mm) with a clay content in the range of 3-20% show an exponential decay for the hydraulic conductivity.
iii
Contents
Summary
vii
Preface
viii
1. Introduction
1
2. Theory
4
2.1. Pore Space Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1. Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2. Porosity and Saturated Water Content . . . . . . . . . . . . . . . . . . . . . 4 2.1.3. Specific Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.4. Irreducible Water Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2. Saturated Hydraulic Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.1. Darcy’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.2. Estimating Hydraulic Conductivity . . . . . . . . . . . . . . . . . . . . . . 7 2.3. Electrical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.1. Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.1.1. Electrical Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.1.2. Spectral Induced Polarization (SIP) . . . . . . . . . . . . . . . . 8 2.3.2. Electrical Properties of Sediments . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.2.1. Electrical Conductivity of Sediments . . . . . . . . . . . . . . . . 8 2.3.2.2. Spectral Induced Polarization (SIP) of Sediments . . . . . . . . . 10 2.4. Nuclear Magnetic Resonance (NMR) . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.1. Nuclear Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.2. NMR Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.2.1. Free Induction Decay (FID) Experiment: Measurement of T ∗2 . . . 16 2.4.2.2. Carr Purcell Meiboom Gill (CPMG) Experiment: Measurement of T 2 17 2.4.2.3. Inversion Recovery (INVREC) Experiment: Measurement of T 1 . 19 2.4.2.4. Pulsed Field Gradient NMR (PFG NMR) . . . . . . . . . . . . . . 20 2.4.2.5. Diffusion Experiment: Measurement of D 0 . . . . . . . . . . . . . 21 2.4.3. Application in (Hydro)Geophysics . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.3.1. Scale Dependency of Hydraulic Conductivity . . . . . . . . . . . 22 2.4.3.2. Relation NMR and Porous Media . . . . . . . . . . . . . . . . . . 23 2.4.4. Pore Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.4.1. Pore Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.4.2. Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.4.3. Hydraulic Conductivity . . . . . . . . . . . . . . . . . . . . . . . 26 3. Material
28
3.1. Sample Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Sample Housing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
28 29 30
Contents 4. Measurement Methods
31
4.1. Pore Space Properties . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Density . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Porosity and Water Content . . . . . . . . . . . . . . . 4.1.3. Specific Surface . . . . . . . . . . . . . . . . . . . . . 4.1.4. Irreducible Water Saturation . . . . . . . . . . . . . . . 4.2. Hydraulic Conductivity . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Falling Head Permeameter . . . . . . . . . . . . . . . . 4.2.1.1. Method and Equipment . . . . . . . . . . . . 4.2.1.2. Post Processing Falling Head Data . . . . . . 4.2.2. Constant Head Permeameter . . . . . . . . . . . . . . . 4.2.2.1. Method and Equipment . . . . . . . . . . . . 4.2.2.2. Post Processing Constant Head . . . . . . . . 4.2.3. PERO Permeameter . . . . . . . . . . . . . . . . . . . 4.3. Electrical Properties . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. Equipment . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Post Processing FDEC Data . . . . . . . . . . . . . . . 4.4. Nuclear Magnetic Resonance (NMR) . . . . . . . . . . . . . . 4.4.1. Equipment and Operation . . . . . . . . . . . . . . . . 4.4.2. Calibration of Equipment . . . . . . . . . . . . . . . . . 4.4.3. Post Processing NMR Data . . . . . . . . . . . . . . . . 4.4.3.1. RI Winfit for Relaxation time constant . . . . 4.4.3.2. RI WinDXP for Relaxation Time Distribution 4.4.3.3. RiDiff for Molecule Diffusion Constant . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
31 31 33 33 34 35 35 35 35 38 38 38 39 41 41 42 42 43 43 44 44 45 46
5.1. Relation Specific Surface and Clay Content . . . . . . . . . . . . . . . . . . . . . . 5.2. Relation Hydraulic Conductivity and Resistivity . . . . . . . . . . . . . . . . . . . . 5.3. Hydraulic Conductivity and NMR properties of Samples . . . . . . . . . . . . . . .
47 48 54
5. Results and Discussion
47
6. Conclusion and Outlook
65
Bibliography
67
Acknowledgements
70
I.
71
Appendices
A.
72
A.1. A.2. A.3. A.4.
Contents of Samples . . . . . . . . . . . . . . . . . . Pore Space Properties of Samples . . . . . . . . . . . Hydraulic Conductivity of samples . . . . . . . . . . . Conversions Between Units for Hydraulic Conductivity
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
72 73 73 74
B.1. PERO Permeameter Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
B.
75
C.
76
C.1. Frequency Dependent Electrical Conductivity Measurements (FDEC) . . . . . . . . C.2. Resistivity versus Clay Content and Specific Surface . . . . . . . . . . . . . . . . . C.3. Resistivity versus Fluid conductivity . . . . . . . . . . . . . . . . . . . . . . . . . .
v
76 80 82
Contents D.
84
D.1. D.2. D.3. D.4. D.5. D.6. D.7.
NMR decay time and amplitude versus volume. . . . NMR Pulse Sequences used with Maran Ultra 2 MHz NMR Relaxation Times . . . . . . . . . . . . . . . . NMR Measurements: T2 relaxation time curves . . . NMR Measurements: T1 relaxation time curves . . . NMR Measurements: T1 relaxation time distributions Relations between NMR and pore space properties .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
84 86 88 89 91 93 95
E.1. List of Abbreviations and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
E.
97
vi
Summary Nuclear magnetic resonance (NMR) is a technique at molecular scale that uses the electromagnetic properties of atomic nuclei to obtain information about molecules. NMR in applied geophysics is used to obtain information about the abundance of water molecules (hydrogen nuclei) and the relation between water and porous media. From the abundance of hydrogen nuclei the porosity (φ ) of saturated pores can be obtained, the NMR properties of water in porous media gives information about pore size. Both are related to hydraulic conductivity (k). The applicability of laboratory NMR to obtain hydrologic parameters as porosity and pore size are investigated. These parameters are used to derive the saturated hydraulic conductivity. NMR is also used at field scale and referred to as surface nuclear magnetic resonance (SNMR). SNMR is a geophysical technique that has proven to be a useful tool for the investigation of hydrologic properties (porosity and hydraulic conductivity) of aquifers. The relation between measurements of SNMR (field scale) and NMR (laboratory scale), however, is uncertain. To obtain an enhanced understanding of SNMR data a comparison is made with laboratory NMR data. To verify the laboratory NMR measurements, pore space properties have been analyzed (specific surface, porosity). Frequency dependent electrical conductivity (FDEC) measurements were carried out to investigate the relation with NMR. Finally hydraulic conductivity measurements were conducted to verify the saturated hydraulic conductivity derived from NMR relaxation time and saturated hydraulic conductivity estimates based on grain size (after Hazen, Kozeny-Carman). Until recently laboratory NMR only focused on consolidated sediments, in this research an attempt is made to investigate unconsolidated sediments. Physical properties of synthetic and natural unconsolidated samples were analyzed: artificial glass pearls, sand mixtures, bore-core samples from a Quaternary environment west of Berlin (Germany) and coarse sand samples with a varying clay content. The presence of clay resulted in a strong influence of all conducted measurements and parameters and is a dominating factor in unconsolidated sediments. The results show that the relation used to obtain hydraulic conductivity from NMR is suitable to predict the saturated hydraulic conductivity for clay and coarse sand samples but not for intermediate grain sizes. A more individual approach with regard to the paramagnetic properties of the material might be needed to allow successful estimations over a wide grain size. The simple Hazen model to estimate the saturated hydraulic conductivity resulted in better estimates of the saturated hydraulic conductivity than the more sophisticated Kozeny-Carman model. The tortuosity used in the saturated hydraulic conductivity estimations of the Kozeny-Carman was taken constant (with a value of 1.5). The model is assumed to give better results when a differentiation in tortuosity is made for different grain sizes. Resistivity information of the FDEC measurements could be used, the phase information of FDEC data that was obtained could not be used because it was too noisy. A comparison between SNMR and NMR showed that there is a difference for fine sand to coarse sand mixtures. Very fine sands and sands with clay show a good agreement.
vii
Preface This MSc thesis has been made at the department of applied geophysics at the Technical University of Berlin, Berlin, Germany. It is not common for hydrology students (officially Soil, Water and Atmosphere studies) of our (sub-)department to make a MSc thesis at a geophysical department. The reason why I made my thesis here is as follows. In the year 2000 I followed geophysical courses at the geophysics department of Utrecht University, Utrecht, the Netherlands. At the end of that year the European Association of Geoscientists & Engineers (EAGE) held a congress in Amsterdam (the Netherlands). An interesting talk was given by Marian Hertrich about surface nuclear magnetic resonance (SNMR). My interest in SNMR became a fact. At the moment I had to think about a topic of my second MSc thesis (end of summer 2002) the Journal of Applied Geophysics came with a ”special issue” on SNMR with guest authors Prof. Yaramanci and L. Guillen, the first being the professor of the department of geophysics at the TU of Berlin. One e-mail to him with the question if there would be a place for a hydrology student from Wageningen (the Netherlands) at his department was enough, the answer is in front of you.
viii
1. Introduction Groundwater constitutes an important component of many water resource systems, supplying water for domestic use, for industry and for agriculture (Bear, 1972). It is also very important for many eco-systems as these eco-systems are closely related to shallow water tables. The production of drinking water becomes more expensive as the quality of raw water decreases, agriculture can be less profitable as irrigation water has a bad quality and eco-systems can be completely distorted when the groundwater quality changes (Valstar, 2001). Still a lot of people do not have access to fresh drinking water of acceptable quality and when no measures are taken this will only become worse in future. To ensure that available fresh water can be distributed in a fair way, insight in the behavior of the dynamics of water is needed. As the water runs in a cycle through different ”sphere’s”, as there are atmosphere, hydrosphere, lithosphere and a kryosphere, knowledge of these different fields is needed to describe this dynamic behavior. To put it another way, water is a connecting component of these different fields and should therefore be investigated in an interdisciplinary manner. Geophysics is an important field of science that can contribute to a better understanding of the hydrological system. The hydrological system is complex and in order to understand and predict its behavior the hydrological system is simplified. This simplification is needed to allow for the modeling of the hydrological processes. In the past thirty years hydrological models have undergone an intensive development. Analog models are replaced by digital models and they became more and more complex. Together with this increase in complexity is the demand for model data. More detailed hydrological models need more detailed information. The most important parameter in groundwater hydrology is the (saturated) hydraulic conductivity (k) of the media the groundwater flows through. This parameter determines how fast (or how slow) water particles can flow through the subsurface for a given potential gradient. This is of great importance for groundwater hydrologists that want to know, for example, how fast a groundwater pollution is transported from A to B (Kooman, 2002). Hydraulic conductivity is generally not homogeneous, i.e. it varies with depth and space. To obtain information about the hydraulic conductivity, field measurements can be performed. For example a so-called pumping test can be applied to retrieve direct information about the hydraulic conductivity (Kruseman et al., 1990). Such tests, unfortunately, are both expensive and time consuming. The obtained information on the other hand is usually of high quality but the hydraulic conductivity is ”lumped” over a large volume. Especially groundwater quality models (models that describe the flow of the fluid and its solutes) require high resolution data. Small variations in hydraulic conductivity can have a severe influence on the flow path followed by the fluid. Therefore it would be interesting to have a method that is cheaper, quicker and provides high resolution data representative for a larger area (Slater and Lesmes, 2002). In geophysics there are methods that fulacquisitionfill the needs mentioned above. One acquisition method provides depth dependent data while another acquisition method provides laterally dependent data, so-called ”mapping” of the subsurface. There are geo-electrical sounding, electromagnetic sounding, georadar and surface nuclear magnetic resonance sounding (SNMR, Yaramanci et al., 1999). Almost all these methods are indirect methods, i.e. physical parameters are measured that can be related to structural parameters (sediment type, porosity, hydraulic conductivity). SNMR however, is (Legchenko et al., 2002) a direct method for detection of free water in the subsurface. The principle NMR is based on the phenomenon that atomic nuclei posses a magnetic moment and a ”spin” (Levitt, 2001). The spin of the nuclei are generally in equilibrium with a static magnetic field. This can either be the earth magnetic field or the permanent magnetic field from a magnet in a NMR spectrometer. The nuclei have a spin that precesses around this static magnetic field. Each nucleus has its own
1
frequency which it can transmit and adsorb electromagnetic energy, the so-called Larmor frequency, Ω0 (Callaghan, 1991). When an oscillating magnetic field with exactly this frequency is transmitted the spin of the nuclei is no longer in equilibrium. After this radio frequency pulse (RF) the spins ”relax” back to their equilibrium state. This produces the NMR signal. This signal provides information about the amount of the nuclei (amplitude) and the direct vicinity of the nuclei as this influences the relaxation rate. When a RF pulse with the Larmor frequency of protons is used, information about water molecules (and/or hydrocarbons) and their surroundings can be obtained. Although SNMR is a geophysical method that is able to detect free water directly it does not obtain direct information about water content or hydraulic conductivity. There is, however, a relationship between hydraulic conductivity and relaxation time and amplitude and water content respectively. With laboratory NMR the same phenomenom is measured though at a smaller scale. The relations between structural parameters derived from surface NMR at field scale and from laboratory NMR at micro scale are uncertain. A part of this MSc research deals with the structural parameters obtained from laboratory NMR only, though a comparison is made with SNMR. It is, as such, an investigation in the interpretation of SNMR data. As already briefly mentioned, (S)NMR is not the only geophysical technique available for the investigation of the subsurface. A method commonly used is electrical resistivity. The resistivity of sediments in the subsurface provide information about the connectivity of pores and electrical properties of the material itself. These electric properties are related to hydraulic conductivity via the specific surface (Sm ) and the surface-to-pore-volume, S por of the material. Electrochemical processes between the pore fluid and the material cause the observed resistivity to be dependent with frequency. The result is a complex (frequency dependent) resistivity, called spectral induced polarization (SIP). To obtain SIP information, frequency dependent electrical conductivity (FDEC) measurements are performed. Information about The structural information obtained by NMR is influenced by mostly the same material characteristics as electrical resistivity. The mechanisms for both methods are different however and the information might be complementary. The relation between NMR, SIP and hydraulic conductivity is the other topic of research in this thesis. The key parameters for hydraulic conductivity, electrical resistivity and NMR are pore space characteristics, i.e. specific surface and porosity. These structural parameters are measured indepently and form the connection between SIP and NMR. Problem Definition
• Deriving geohydrological parameters (hydraulic conductivity, porosity) from laboratory measurements (low magnetic field strength NMR and FDEC) • For synthetic samples, unconsolidated sediments as well as field samples • To measure the saturated hydraulic conductivity directly, and verify obtained estimates Objectives Until recently laboratory NMR measurements mainly focused on consolidated sediments (mostly sandstones) and only a few attempts have been made to investigate unconsolidated sediments. A comparison between laboratory NMR, FDEC, pore space properties and saturated hydraulic conductivity in a broad sense as in this thesis is still quite unique. The main objectives of this work are therefore:
• Investigate the application of Laboratory NMR to estimate hydraulic conductivity of unconsolidated sediments • Get insight in data provided by NMR measurements in the field (SNMR) • Investigate the applicability of modern geophysical method in groundwater hydrology
2
The thesis has the following content:
In this chapter the theory that is used for the measurements is outlined. The theory per method is described in different sections. The order of the sections has been arranged in such a way that the principles that are used in other measurements are explained first. The part on NMR is extensively elaborated as this is a quite new method for hydrologists. chapter 2
This chapter deals with the type of samples used and how these were prepared. The chapter only deals with the sample and the used material, as the method and equipment used for the measurements is explained in chapters 2 and 4, respectively. chapter 3
The same structure as for the chapter 2 has been applied to make it easier to go ”back and through” while reading.
chapter 4
The results are shown and discussed together because these are tightly linked. Conclusions are drawn in chapter 6 and recommendations for further research are given.
chapter 5 and 6
3
2. Theory In this chapter the concepts and definitions used to obtain the saturated hydraulic conductivity from pore space properties, FDEC and NMR measurements are explained. It provides the theoretical framework so that later on (Chapter 5) comparisons between the performed measurements and derived material properties can be made.
2.1. Pore Space Properties In this work pore space properties are regarded as a general term to relate properties from the solid framework of grains (matrix) to the non-solid matrix, i.e. pore space. The properties of the medium (matrix) determine the structure of the pore space, and as the pore space the characteristics of the medium in terms of hydraulic conductivity, they are two sides of the same coin. They can be used to estimate the hydraulic conductivity. The theory of how the hydraulic conductivity can be estimated is outlined in Subsection 2.2.2. 2.1.1. Density
Density, denoted by ρ , is defined as the quotient of the mass m and the volume V of a material:
ρ=
m [kg/m3 ] V
(2.1)
In this research two different densities are used. • ρ , bulk density (or total density): the density of the considered sediment volume (including pores). • ρmatrix , matrix density: density of an individual sediment component (i.e. quartz, or glass pearl), thus without pores.
ρmatrix it is thus a material property of the building blocks (grains) that together build the skeleton of grains (matrix). 2.1.2. Porosity and Saturated Water Content
The porosity φ is the fraction of the total volume occupied by pores. Pores are local enlargements in a pore-space system that provides the volume available for fluid and/or gas storage. The pores can be connected to each other by smaller spaces called pore throats (Schön, 1996). Two kinds of porosity are defined here: • Total porosity, φtotal is the ratio of the volume of the pore space V pore to the total volume (volume of matrix and pores) also called bulk volume, Vbulk of the sample
φtotal =
Vpore [−] Vbulk
(2.2)
and with the volume of the matrix (volume of grains), Vmatrix the total porosity can be written as:
φtotal = 1 −
4
Vmatrix [−] Vbulk
(2.3)
2.2. SATURATED HYDRAULIC CONDUCTIVITY • Effective porosity, φe f f : is the ratio of the volume of the pore space that is interconnected (volume available for fluid flow, Ve f f ) to the bulk volume Vbulk of the sample
φe f f =
Ve f f Vmatrix +Visolated = 1− [−] Vbulk Vbulk
(2.4)
Where Visolated is the pore volume that is isolated from the rest of the pore space (not connected with other pores). The volumetric water content θ , is the ratio of the volume of water Vwater to the pore volume Vpore
θ=
Vwater Vwater = [−] Vtotal Vmatrix +Vpore
(2.5)
2.1.3. Specific Surface
Specific surface S [1/µ m] is defined as the ratio of surface (m 2 ) of the pores to the pore volume (m3 ) or sediment mass (m2 /kg). Several different types of specific surface are used, i.e. dependent on which property (mass of volume of rock) it is related with (Schön, 1996): • related to total rock volume:Stotal • related to pore volume (also surface-to-pore-volume, S/V): S por S por =
S Vpore
=
m2 1 = m3 m
(2.6)
• related to volume of the solid matrix: S mass • related to mass of the matrix: Sm Stotal = φtotal S por = (1 − φtotal ) Sm [1/µ m] Sm =
Smass [m2 /g] ρmatrix
(2.7) (2.8)
2.1.4. Irreducible Water Volume
Irreducible water volume Swi , is the part of the water in a pore that stays bounded to the pore wall surface when a certain underpressure (centrifuging) is applied. It is also reffered to as ”capillary bound water”. Clay is the main factor in binding water molecules. It can be used to predict the intrinsic permeability (Subsection 2.4.4.3).
2.2. Saturated Hydraulic Conductivity Saturated hydraulic conductivity is determined by the pore space properties (porosity and specific surface) and as such could be listed under this term, but because saturated hydraulic conductivity is the most important parameter of this thesis it has its own section. It is an indicator that characterizes the medium (pore space) it flows through. The word ’saturated’ indicates that the pore space is completely filled with fluid (water), i.e. is saturated. Fluid flow under unsaturated conditions is logically referred to as unsaturated hydraulic conductivity. That is however a special case, and a field of science on its own. This research only deals with saturated hydraulic conductivity. Therefore, from here on, when hydraulic conductivity is discussed saturated fluid flow is meant.
5
2.2. SATURATED HYDRAULIC CONDUCTIVITY 2.2.1. Darcy’s law
Hydraulic conductivity is the ability of a fluid to flow through (interconnected) pore space. A porous sediment is therefore regarded as a porous medium. Flow through porous media is given by Darcy’s law (Bear, 1972): v = −k∇H [m/d]
Where:
(2.9)
v [m3 /m2 /d] = volume flux density k [m/d] = hydraulic conductivity ∇H [−]= hydraulic head gradient The permeability k is given by the following relation: k=
ρg κ [m2 ] η
(2.10)
Where:
κ [m2 ] = intrinsic permeability ρ [kg/m3 ] = volume mass density η [kg/ms] = dynamic viscosity κ is the so-called intrinsic permeability and the properties of the pore geometry. ρηg characterizes the properties of the fluid. The dimension of κ in SI-units is m 2 . It is the intrinsic permeability that the petroleum engineers and petrophysicists refer to as the conductivity, they use the (non-SI) unit Darcy. 1 Darcy = 9, 8692.10−13 m2 ≈1 µ m2 . See Table A.3 and Table A.5 in Appendix A.4 for conversions between different units. The volume flux density, v is also-called filter velocity. It is not the true velocity of the water molecules but a volume transported water per surface. To calculate the true velocity of the water molecules, the filter velocity has to be divided by the effective porosity, φ e f f (Bear, 1972): vtrue =
v
φe f f
[m/d]
(2.11)
In practical situations equation 2.9 is written in its finite form: ∆H (2.12) ∆X Where ∆H [m] is the hydraulic head difference over a distance ∆X [m]. Hydraulic head gives the amount of potential energy per weight at a point (i.e. point A). It has two components, one is the height of the point above a reference level, z A , the other is the pressure height, h Pa (see Figure 2.1) v = −k
HA = zA +
PA = zA + hPa [m] ρwater g
6
(2.13)
2.3. ELECTRICAL PROPERTIES
hPa HA zA Reference level, z=0 Figure 2.1.: Definition of hydraulic head. 2.2.2. Estimating Hydraulic Conductivity
Instead of measuring the hydraulic conductivity, it can be estimated. This can be done if some characteristics of the porous media (see Section 2.1) are known. For this purpose several different models are developed. In this research two of them are used: • Model of Kozeny-Carman (after Schön, 1996)
κ=
φe f f [d] 2 T S2por
(2.14)
Where Ttor is the tortuosity. Note that the intrinsic permeability, κ has to be converted from darcy, d to hydraulic conductivity, k [m/d] (see Table A.5). • model of Hazen (Hölting, 1989)
2 k = C d10 [m/s]
(2.15)
T (and T= temperature in degrees Celsius) is an empirical pre-factor. Equation Where C = 0.7 +0.03 86.4 , where d10 and d60 are percentiles of the grain size 2.15 is only applicable if U < 5 with U = dd60 10 distribution. d10 and d60 mean the fraction of the grain size distribution left of 10%, 60% of the distribution respectively, the grain size distribution is plotted on a logarithmic axis.
2.3. Electrical Properties 2.3.1. Basics 2.3.1.1. Electrical Resistivity
The equation that describes electrical resistivity is Ohm’s Law: I=
U [A] R
(2.16)
Where I is the current, U [V ] the voltage and R [Ω] the electrical resistivity. The resistivity of homogeneous material can be calculated as: R=ρ
l A
(2.17)
Where l is the length of the sample [m], ρ res the resistivity [Ωm] and A [m2 ] is the surface. When Ohm’s law (equation 2.16) is substituted in equation 2.17 and rewritten, the specific resistivity can be determined by measuring the voltage and the current:
ρ=
UA [Ωm] I l
7
(2.18)
2.3. ELECTRICAL PROPERTIES 2.3.1.2. Spectral Induced Polarization (SIP)
Ohm’s law is suitable to describe electrical resistivity for simple situations, i.e. electric circuits without capacitors. For situations where capacitive effects play a role, however, the resistivity is more complex, figurly and literally. In such a case the resistivity has not only a real part but also an imaginary part, e.g. it is expressed as a complex number. The total current density in the frequency domain can be written as (Jackson, 1975): Jtotal = Jcond + J pol = (σ + jωε ) E [A/m2 ]
(2.19)
Where ω is the angular frequency (ω = 2π f ), E is the electric field strength [V /m], ε is the dielectric permittivity [F/m] and σ [S/m ] is the conductivity (reciprocal of resistivity) . With the definition of the complex spectral induced polarization (SIP) the electrical conductivity is separated in a real and imaginary part:
σ ∗ (ω ) = σ (ω ) + jωε (ω ) = σ 0 (ω ) + jσ 00 (ω ) [S/m]
(2.20)
In general the imaginary part of the complex conductivity represents the polarization. The real part of the complex conductivity consists of both conductive as polarization effects. From the complex conductivity the amplitude and phase of the resistivity can be calculated: Amplitude : |ρ (ω )| = and
Phase :φ (ω ) = arctan
n
σ 0 (ω )
2
+ σ 00 (ω )
2 o− 21
[Ωm]
−σ 00 (ω ) Imaginary part {ρ } = arctan 0 [rad] Real part {ρ } σ (ω )
(2.21)
(2.22)
Note that electrical conductivity and electrical resistivity are equivalent, i.e. one is the reciprocal of the other. In fact is does not matter what parameter is used. In some cases however it is more common to talk about resistivity, in others conductivity. A material can have a low ”resistivity” or a high ”conductivity”. 2.3.2. Electrical Properties of Sediments
Electric properties of sediments are interesting as the electric current is conducted by the pores and surface of the pore walls. Both pore space properties are strongly related with the hydraulic conductivity. What kind of processes influence the electric conductivity of sediments is outlined in the following subsection. 2.3.2.1. Electrical Conductivity of Sediments
In geophysics four different mechanisms of electrical conductivity are recognized (Schön, 1996): 1. electrical (metallic) conductivity: charge transport by free electrons in mineral (i.e. iron) 2. electrolytic conductivity: charge transport by mobile ions in the pore fluid 3. interface conductivity: minerals act as solid electrolyte 4. dielectric polarization: charge transport by polarization current at high frequency. The measured conductivity can (and in most cases it will) be a combination of above mechanisms. In this research only a combination of the second and third mechanism are active.
8
2.3. ELECTRICAL PROPERTIES Sediment Conductivity
Sediment conductivity can be modeled as a parallel conductor system with an electrolytic and an interface conductivity component (Rink and Schopper, 1973):
σsediment = a φemf f S−n σelectrolytic + σinter f ace [S/m]
(2.23)
Where:
σsediment [S/m] = the conductivity of the sediment φemf f [-] = the effective porosity to the power of m (cementation exponent) S−n [-] = the saturation to the power of −n (saturation exponent)
σelectrolytic [S/m] = the conductivity of the fluid σinter f ace [S/m] = the conductivity of the interface or ”double-layer” conductivity a [-] = proportionality factor Electrolytic Conductivity
The electrolytic conductivity of the sediment has been empirically described by Archie (Archie, 1942): −n σsediment = a φe−m σelectrolytic ff S
(2.24)
With the saturation factor S (for saturated samples is equal to 1) and n is the saturation exponent. The factor a is a proportionality factor (0.5 < a < 1), m the cementation exponent ranging from 1.3 for unconsolidated sediments (sands) to 2.4 for consolidated sediments (sandstones). For saturated sediments (S = 1) without interface conductivity, this factor relates the sediment conductivity and the electrolytic conductivity: σelectrolytic σsediment = (2.25) F Where F [-] is called the ”formation resistivity factor”. Interface Conductivity
The interface conductivity represents the conductivity contributed by the ”double-layer”. An electric double layer is formed by clay minerals. Clay minerals are negatively charged and have therefore the capacity to bind cations. When the sediment containing clay minerals is saturated with water, these cations can go into solution but are loosely held to the (negatively charged) surface. This layer (interface) can act both as a conductor and a capacitor (Rink and Schopper, 1973). The determination of the interface conductivity term from the total sediment conductivity based on equation 2.23 needs measurements made with different fluid conductivities σ f luid . However it can be roughly said (Schön, 1996): • real part of conductivity is influenced by the volume (electrolytic) and interface properties • imaginary part of conductivity is influenced by the interface effects alone The most commonly used model to describe the complex conductivity (see Subsection 2.3.1.2) is the Cole-Cole model (after Pelton et al., 1978): 1 ρ (ω ) = R 1 − ζ 1 − [Ωm] (2.26) 1 + (iωτ )c
9
2.3. ELECTRICAL PROPERTIES
Figure 2.2.: Cole-Cole diagram in time domain (upper part) and frequency domain (lower part). After (Kretzschmar, 2001) Where:
ζ [-] = chargeability c [-] = the frequency dependence constant
τ [s] = the relaxation time constant i=
√ −1
In Figure 2.2 a typical Cole-Cole diagram is shown. On the transition of the amplitude from high to low there is a minimum in the phase.
2.3.2.2. Spectral Induced Polarization (SIP) of Sediments
The SIP discussed in Subsection 2.3.1.2 can be seen as a current-stimulated electrical phenomenon observed as a delayed voltage response in earth materials (Schön, 1996). Induced polarization has two different underlying physical/chemical mechanisms • Electrode Polarization or metallic polarization (i.e. pyrite) • Membrane Polarization or boundary layer polarization (i.e. clay) Electrode polarization occurs in an electric field if an electronic conducting mineral (i.e. pyrite) is in contact with an ionic conducting pore fluid. There is a current flow from ionic to electronic (or metallic) and vice versa. This is illustrated in Figure 2.3A. The current flows across the interface encounters a certain resistivity. The cations and anions in the electric field move in opposite directions but the presence of the mineral forms a barrier (Figure 2.3B) against this flow because electrons cannot go into the solution and ions cannot go into the mineral. The charges are exchanged by electrochemical (oxidation, reduction) processes. This gives rise to clouds of ions with different polarities. In the mineral charges with opposite polarity are produced. Electric charge is thus accumulated and when the electric field is switched off the charges diffuse away (Schön, 1996).
10
2.3. ELECTRICAL PROPERTIES
+ A
interlayer
electrolyte -
+ + + + +
cation
+ B
mineral grain
anion
Figure 2.3.: Mechanism of Electrode Polarization. Figure A, electrolyte without mineral. Figure B, electrolyte with mineral (after Schön, 1996). Membrane polarization, the second physical-chemical mechanism for induced polarization, is important for sediments that contain a few percent of clay throughout the matrix. Clay minerals can form a so-called double layer, i.e. a diffuse cloud of cations in the vicinity of a clay surface (Figure 2.4A). Under influence of an electric field , cations can easily pass through this positively charged cloud but forms a barrier for anions (Figure 2.4B). This double layer acts as an ion-selective membrane. After the electric field is switched off all charges diffuse away again (see Figure 2.4). The resulting ion concentration gradient opposes the current flow, thereby reducing the overall mobility of ions. This effect is frequency dependent as for low frequencies this effect is stronger than for high frequencies in terms of ion diffusion time.
- - - --
-
A
- - - - -
clay electrolyte
-
anions blocked
- - -- - -
-
+ B
- - - - -
cation anion
Figure 2.4.: Mechanism of Membrane Polarization, Figure A, without electric field. Figure B, with electric field. (after Schön, 1996). The effect of membrane polarization also depends on the pore size. The electric double layer is thin compared to the pore volume for large pores. Only a small percentage of the ion motion will thus be influenced. For a small pore or the part pore throats connecting two pores the electric double-layer can touch each other and be an effective barrier. This model is referred to as ”passive and active pore sections” (Schön, 1996, see Figure 2.5). passive zones P -
-
- -
- -
-
-
- -
- - -
-
-
- - - - - -
- -
- -
- - - - - - - - -
-
-
-
-
-
-
-
- -
-
-
-
-
-
- -
-
cation anion Active zones
Figure 2.5.: Polarization by constrictivity of pores (after Schön, 1996).
11
2.4. NUCLEAR MAGNETIC RESONANCE (NMR) In this thesis the membrane polarization is the reason for the observed SIP effect. In Figure 2.6 it is shown which kind of parameters can be derived from FDEC measurements.
Spectral Induced Polarization (SIP) Frequency dependent Amplitude Resistivity ( r(w) )
Frequency dependent Phase (f(w) )
Real part Imaginary part of conductivity ( s ' (w ) ) of conductivity ( s " (w ) ) Capacitive interface contribution
- Surface-to-pore-volume ratio (S/V) - water saturation (S)
Conductive interface contribution - Formation factor (F) - Salinity - Water saturation (S)
Electrolytic conductivity (s electrolytic) ARCHIE component
Figure 2.6.: Flow chart to derive pore and fluid properties from SIP data (after Schön, 1996) From the FDEC measurements at the top of the flow chart the amplitude, ρ (ω ) and phase, φ (ω ) of the resistivity can be calculated (Equation 2.21 and 2.22). The amplitude represents the real part of the resistivity, the phase the imaginary part. The real part of the resistivity is used to calculate the sediment conductivity, σsediment (Equation 2.24), the formation resistivity factor, F (2.25), the water saturation, S, and the salinity (Equation 2.23). The imaginary part which exist out of a capacitive conductive interface (membrane polarization) and a conductive interface contribution (conducting ”double layer”). From this part the water saturation, S and information on the specific surface can be obtained.
2.4. Nuclear Magnetic Resonance (NMR) 2.4.1. Nuclear Magnetism
In order to understand nuclear magnetic resonance (NMR) it is important to have knowledge about some basic processes. The basics of NMR is formed by the theory of electromagnetics. The electromagnetic field is described by two fields. The electric field E, which interacts with electric charges and the magnetic field B which interacts with magnetic moments (Levitt, 2001). All atomic particles have a magnetic moment which means that they can interact with magnetic fields. This interaction can be expressed as the magnetic moment µ [A m 2 ]. The amount of magnetic energy in this interaction is related to the magnetic field strength B [A m] and can be written as: Emagnetic = −µ · B [J]
12
(2.27)
2.4. NUCLEAR MAGNETIC RESONANCE (NMR) When the vectors µ and B both point in the same direction the magnetic energy is the lowest. When oriented opposite to each other the magnetic energy is maximal. The magnetic moment can be permanent (paramagnetic material) or induced (diamagnetic material). Most of the substances fall in the second class and only have a magnetic moment when an external magnetic field is applied. The development of such an induced field takes some time and the strength of the field is related to the applied (external) magnetic field and the “willingness” (susceptibility) of a material to develop a magnetic field. They are related to each other as:
µinduced = µo−1V χ B [A/m2 ]
(2.28)
Where µo = 4π · 10−7 [H/m] the permittivity of vacuum, V the volume and χ [-] the susceptibility of the material. A positive susceptibility indicates paramagnetic materials while a negative susceptibility is characteristic for diamagnetic material. The explanation above describes macroscopic magnetism. It describes the effect but it does not give an explanation of where this magnetism comes from. To get a more thorough understanding a closer look on the matter is needed. Three sources of magnetism can be identified: 1. circulation of electric current 2. magnetic moments of the electron 3. magnetic moments of the atomic nuclei With a decreasing influence from top to bottom. The first source is known from induction theory. A moving electric current induces a magnetic field and the other way round. The other two sources “simply” are intrinsic properties of electrons and nuclei. Another intrinsic property of electrons and nuclei that becomes important at this point is spin angular momentum. Both electrons and nuclei thus have a permanent magnetism. This is illustrated in Figure 2.7.
Figure 2.7.: Magnetic moments of electrons and nuclei, (from Levitt, 2001) The following relation exists between the magnetic momentum and the spin angular momentum:
µˆ = γ Sˆ [A m2 ]
(2.29)
Where γ [Hz/T ] the gyromagnetic ratio for protons (γ proton = 42.54597 [MHz/T ]) and Sˆ [A m2 ] the spin angular momentum. The “hats” in equation 2.29 indicate quantum mechanical operators (Levitt, 2001). Stable atomic nuclei posses both angular momentum (spin) as magnetic momentum. In quantum mechanics, angular momentum is quantized. It is quantitized in integer or half-integer units (I) of
13
2.4. NUCLEAR MAGNETIC RESONANCE (NMR) h¯ = 2hπ (where h is the Planck constant, h = 6.62608 · 10 −34 [J s]), depending on whether the number of constituent nucleons is even or odd (Callaghan, 1991). Quantum mechanics tells us that a nucleus of spin I will have 2 I+1 possible related energy levels. In the absence of an external magnetic field these energy levels are equal. If an external magnetic field is applied, the energy is split (Zeeman splitting) and the energy level of the spin orientations differs slightly. For most particles and nuclei the magnetic moment points in the same direction as the spin angular momentum (γ > 0), some may have it pointed in the opposite direction (γ < 0). Particles with spin, for example, proton nuclei have an angular momentum vector pointing in any direction in space. This is called the spin polarization axis. Although this quantization of angular momentum, the direction of the spin polarization axis can be in all possible directions. Thus will be randomly distributed (each with its own spin orientation, see Figure 2.8).
Figure 2.8.: Randomly distributed spin polarizations (from Levitt, 2001) When an external magnetic field is applied to the nuclei (earth magnetic field or magnet from a NMR spectrometer) the magnetic moment of the nuclei tries to align with this field, often noted as B0 [A/m]. Assume this field points in the positive z-axis. As the nuclei posses not only magnetic momentum but also spin angular momentum, it does not align completely, but instead moves around the applied magnetic field (see Figure 2.9). This movement can be compared with the spinning of a top and is called precession. The angle on which the spins process is dependent on the “starting” position of the nuclei. As the nuclei are randomly distributed, the precession also will be; showing the complete range from zero degree to -180 degree (see Figure 2.10). Applied Magnetic Field (B0)
B0
Precessional Orbit
Spin Precession
Figure 2.9.: Spin precession (after Levitt, 2001).
14
2.4. NUCLEAR MAGNETIC RESONANCE (NMR)
B0
Figure 2.10.: Precession of spins might have all possible angles; from 0 degree to -180 degree. In this figure only a few possibilities are shown (after Levitt, 2001). The frequency with which this precession takes place is dependent on the strength of the magnetic field and the gyromagnetic ratio and is called the Larmor frequency, Ω 0 : Ω0 = −γ B0 [Hz]
(2.30)
As all nuclei own a specific and unique gyromagnetic ratio they also have a distinct Larmor frequency given a certain magnetic field. At this point the Resonance term from NMR may become clear. It is the Larmor frequency at which the nuclei will interact (resonate) with an external applied (oscillating) magnetic field. As discussed before the spin polarizations can point in any possible direction in space when no external magnetic field is applied. If a magnetic field is applied the spins start to precess around this field. Besides the spins of the nuclei there are other processes active. For example translational motions of (water) molecules in space and rotational motion of the molecules around their axis occur. This gives rise to small and fast fluctuations of the local magnetic field of the nuclei. The magnetic moment of the nuclei varies in amplitude and direction with time. Although these processes might be violent and a lot of collisions between molecules and atoms may take place, the precessional motion and direction of the spins is almost undisturbed (Levitt, 2001). Although this effect is only small it is this process that allows nuclear magnetism to be observed! Because of these processes the atomic nuclei perform both precessional motion as well as a “wandering” motion. This wandering motion is not completely random (because of a finite temperature) and it slightly favorites spins with lower magnetic energy above spins with higher magnetic energy. As magnetic energy is related to the spin orientation, spins that are aligned parallel to the magnetic field have a higher probability to exist. This means that a macroscopic magnetization vector can develop and can be observed (with a NMR spectrometer). After the secondary external magnetic field is turned off the processes described above goes on, however due to influence of the nuclei on each other the coherence of the spin polarizations is lost with time. This process is called relaxation because the spins return to their thermal equilibrium (aligned around the earth magnetic field). The time it takes to develop (or decline) the macroscopic magnetization vector is approximately exponential and has the form (Callaghan, 1991): −t
Where:
nuc Mznuc = Meq (2 − e T1 ) [A/m]
Mznuc [A/m] = the macroscopic magnetization vector in direction of main magnetic field (B 0 ) nuc [A/m] = macroscopic magnetization vector at thermal equilibrium Meq
15
(2.31)
2.4. NUCLEAR MAGNETIC RESONANCE (NMR) T1 = Longitudinal relaxation time constant T1 is referred to as longitudinal relaxation time constant or spin-lattice relaxation time constant. Longitudinal indicates magnetization develops in the same direction as the magnetic field. The term relaxation is thus used for the development as as well the decline of the longitudinal magnetization. The longitudinal magnetization, however, is small and almost undetectable. Therefore a different approach is needed to observe the macroscopic magnetization vector. Instead of measuring the magnetization along the field (parallel to main field) the magnetization perpendicular to the field is measured. With the help of a secondary oscillating magnetic field (radio frequency pulse, RF pulse) the longitudinal magnetization vector is flipped π /2 radians around the x-axis. This results in a spin polarization in the negative y-axis (see Figure 2.11).
Figure 2.11.: Rotation of the spin-polarization around the x-axis ( from Levitt, 2001). The magnetization vector that is now perpendicular to the main field (longitudinal magnetization) is called transverse magnetization. As the RF pulse is turned off all spins start to precess. As all spins are polarized in the same direction (- y-axis) the macroscopic magnetization vector precesses in the xy plane with the Larmor frequency (equation 2.30). As discussed earlier the individual spins loose coherence with each other because of locally slightly different settings. This results in the decay of the transverse magnetization vector with time. The relaxation time constant of this decay is called transverse relaxation time constant or spin-spin relaxation time constant and noted T2 . The transverse relaxation time is shorter than or equal to the longitudinal relaxation time (T2 ≤ T1 ). There is, however, a complicating factor for relaxation time constants. In the above it is assumed that all spins experience the same magnetic field on average. Magnetic field gradients in the sample itself and a slightly inhomogeneous external magnetic field (magnet of NMR spectrometer) result in a quicker dephasing of the spins than expected. In practice not the T2 decay time constant is measured but the T2∗ relaxation time constant, with the relation: T2∗ ≤ T2 ≤ T1 . The T2 relaxation time constant is thus masked by inhomogeneous magnetic fields and magnetic field gradients. 2.4.2. NMR Spectroscopy
The theory of how to obtain the parameters T2∗ , T2 and T1 will now be explained. 2.4.2.1. Free Induction Decay (FID) Experiment: Measurement of T∗2
To start with the simplest and quickest measurement, the recording of T2∗ . For the measurement of T2∗ a so-called Free Induction Decay (FID) experiment is performed. At a certain time a RF pulse
16
2.4. NUCLEAR MAGNETIC RESONANCE (NMR) is transmitted with a time length in such a way that the macroscopic magnetization vector is rotated 90 degrees or π /2 radians (Section 2.4.1). It then points in the -y-axis (see Figure 2.11). The pulse sequence itself is given in Figure 2.12.
ð/2 *
T2
t
NMR Signal RF Pulse Figure 2.12.: FID decay, NMR signal after a RF pulse of π /2. The envelope of the decline is the T2∗ relaxation time constant (after Levitt, 2001). Because the T2∗ relaxation time constant is influenced by internal magnetic gradients or external magnetic field inhomogeneities it includes an indication on the strength of these effects. To a lower extent it includes information of the true transverse decay time constant, T2 . To overcome this problem and to be able to measure the true T2 relaxation time constant a special pulse sequence has been designed (Carr Purcell Meiboom Gill experiment). 2.4.2.2. Carr Purcell Meiboom Gill (CPMG) Experiment: Measurement of T2
The Carr Purcell Meiboom gill (CPMG) pulse sequence is also known as ’spin echo pulse sequence’ (Abragam, 1961). After a π /2 RF pulse first there is a T2∗ decay, then after a π RF pulse an echo is formed. The π RF pulses are repeated until the amplitude of the signal is in noise level (see Figure 2.13).
T2 ð
ð
ð/2
t T2
*
Echo 1
Echo 2
t
Figure 2.13.: Spin Echo Pulse Sequence (CPMG).
17
2.4. NUCLEAR MAGNETIC RESONANCE (NMR) Figure 2.14 shows the precession of the magnetization vector during the first τ /2 interval (M 3 ). Because the spins loose coherence with each other, the macroscopic magnetization vector becomes smaller with time.
Figure 2.14.: Precession of the magnetization vector during the first τ /2 interval ( from Levitt, 2001). After the τ /2 interval a second RF pulse is given but this time that turns the magnetization vector 180 degrees or π radians around the y-axis (M4 in Figure 2.11).
Figure 2.15.: Rotation of the magnetization vector by the π y RF pulse (from Levitt, 2001). After the rotation of the magnetization vector the signal is detected after another τ /2 interval time. This exactly compensates the precession during the two pulses, leading to a magnetization vector exactly along the negative y-axis at the begin of the measurement (M 5 in Figure 2.16).
Figure 2.16.: Precession of the magnetization vector during the second τ /2 interval (from Levitt, 2001).
18
2.4. NUCLEAR MAGNETIC RESONANCE (NMR) This process is referred to as refocusing. To understand the effect of the inhomogeneous external magnetic field, each point in space is associated with its own magnetization vector, each with its own precessing frequency. During the decay in the first τ /2 interval the magnetization vectors spread out. This process can be compared to a marathon. In a marathon each runner runs with his or her own velocity to the finish line. Therefore the field of runners is spreaded along the track (see Figure 2.17).
Figure 2.17.: Spreading of the magnetization vectors (from Levitt, 2001). Each magnetization vector is rotated by the π y RF pulse (around the y-axis). This can be interpreted as the replacement of the finish line to the side opposed of the direction the runners run to. The fastest runners are now the most behind and the slowest are now the leaders. In the period after the second RF pulse the magnetization vectors are refocused so that at the begin of the recording time of the experiment all spins point in the negative y-axis again. In this period the fastest runners catch up with the slowest and they all finish at the same time 1 (see Figure 2.18).
Figure 2.18.: Refocusing of the magnetization vectors during the second τ /2 interval (from Levitt, 2001). The result of CPMG pulse sequence on the observed NMR signal is an echo. 2.4.2.3. Inversion Recovery (INVREC) Experiment: Measurement of T1
The experiment for measuring T1 is comparable to the CPMG experiment but with reversed RF pulses. First a RF pulse of 180 degrees or πx radians (rotation around x-axis) is given and after a time τ i the second RF pulse of 90 degrees or πx /2 is given (see Figure 2.19). 1 This
is only completely true when the magnetic field doesn’t change with time
19
2.4. NUCLEAR MAGNETIC RESONANCE (NMR)
(ð/2)x
ðx
t t Figure 2.19.: Inversion-Recovery pulse sequence (after Levitt, 2001). The first pulse sequence is applied to get an inverted spin polarization population. During the interval τ the spins relax back to their (thermal) equilibrium. To check the progress of their relaxation the second RF pulse is given. This signal converts the differences in spin populations into coherences which gives an NMR signal. As the observed signal is a function of the time between the pulses τ and the time after the last pulse, this measurements is repeated for a range of RF pulse lengths τ i . For long τ times the spins will be completely relaxed. This inversion-recovery sequence results in Figure 2.20 and from this decay the longitudinal relaxation time constant, T1 can be obtained.
Pulse Lengths (ti)
a(t)
t T1
Figure 2.20.: Inversion-recovery experiment. NMR signal amplitude as function of different pulse lengths (after Levitt, 2001). 2.4.2.4. Pulsed Field Gradient NMR (PFG NMR)
In the theory to apply NMR to porous media some assumptions are made (see Section 2.4.4). In order to verify these assumptions a special technique of NMR is used. The measurements are formed by pulsed field gradient NMR (PFG NMR). This technique applies a magnetic field gradient on the sample. With a series of RF pulses the sample is probed. Because of the magnetic field gradient the hydrogen atoms will diffuse in the direction of diminishing gradient. The solid structures in porous media effectively provide barriers for self-diffusion. The pulses are separated by a short time period ∆. This results in a net phase shift Φ of the NMR signal (Van As and van Dusschoten, 1997) Φ = γδ gR
(2.32)
If a number of ensembles of spins contribute to the NMR signal the observed signal S is given by (Van As and van Dusschoten, 1997) S=
Z Z
p(r0 )P(r0 |r0 + R, ∆)exp(iγδ gR)d(r0 + R)dr0
(2.33)
Where p(r0 )is the probability to find a spin at position r 0 at time zero, P(r0 |r0 +R, ∆) is the conditional probability that a spin originally at r 0 at time zero (the time of the first magnetic field gradient pulse)
20
2.4. NUCLEAR MAGNETIC RESONANCE (NMR) will be at position r0 + R at t = ∆ of the second pulse and g is the strength of the magnetic field gradient. The integration is over all starting and end positions. From equation 2.33 it is clear that a random distribution of displacements with respect to g does not result in a net phase shifted signal, but in a decrease of the signal amplitude. This will be the case for e.g. diffusion and dispersive flow perpendicular to the net flow direction. In contrast, net flow will result in a net phase shift. By varying the time between the gradient pulses (∆) the displacement can be followed as a function of observation time, allowing tracing of the distance over which the spins can displace (Van As and van Dusschoten, 1997). This principle is used in a so-called diffusion experiment to determine the molecular diffusion constant, D0 . 2.4.2.5. Diffusion Experiment: Measurement of D0
Diffusion constant measurements require the use of a magnetic field gradient to allow for the ”labeling” of the spins (Section 2.4.2.4). As explained in Section 2.4.2.4 this magnetic field gradient is not permanently present but with pulses. First a π /2 RF pulse is applied to bring the spin polarizations in the plane transverse to the permanent magnetic field, just as in the other pulse sequences discussed earlier (Sections 2.4.2.1 to 2.4.2.3). Then a first gradient pulse is applied that labels the spins. Now the spins are labeled and they are rotated again π /2 radians by a π /2 RF pulse. After that there is a waiting period (∆) for the water molecules (protons) to diffuse . After a waiting period ∆ a second gradient RF pulse is applied that ”reads” the position of the protons. Now a so-called stimulated echo is created (see Figure 2.21).
D
(ð/2)x d1
Stimulated echo
(ð/2)x d2
(ð/2)x g1
g2
t
1st gradient pulse
2nd gradient pulse
D
(ð/2)x d1
(ð/2)x d2
(ð/2)x
Stimulated echo
t
D
(ð/2)x d1
(ð/2)x
(ð/2)x d2
Stimulated echo
t
Figure 2.21.: Diffusion Pulse Sequence with different pulse durations to record a stimulated echo. The form of the stimulated echo graph is dependent on the diffusion distance (displacement) of the protons, the time (∆) between the two gradient RF pulses, the duration of the RF pulses (δ ) and
21
2.4. NUCLEAR MAGNETIC RESONANCE (NMR) the strength of the gradient RF pulses (g). The stimulated echo is attenuated when the gradient RF pulse length is varied, this effect is used to calculate the molecular diffusion coefficient (see Section 4.4.3.3). 2.4.3. Application in (Hydro)Geophysics
The mechanisms that play a role in NMR relaxation offer information on the T2∗ , T2 and T1 times which properties of porous media might be derived from. Porous media have a severe effect on NMR relaxation which deviates strongly from neat fluids. Several mechanisms play a role and provide the tools to obtain structural parameters (porosity, pore size (distribution) and hydraulic conductivity, Kenyon (1997)). 2.4.3.1. Scale Dependency of Hydraulic Conductivity
As NMR is a technique based on a molecular level it provides information at a microscopic scale. A porous medium investigated at a microscopic scale shows a complex morphology of pores and surface boundaries of grains (unconsolidated sediments) and/or crystals (metamorphic sediments). To describe the fluid flow through at this microscopic level, although principle possible, it is not realistic, because: 1) the detailed information at a microscopic scale (for a large volume) of a porous medium is unknown. 2) if the information would be available the boundary values (physical boundaries of solid interface) are to complex to translate into a model able to calculate fluid flow at a large scale (Koopmans, 1999). Therfore a different approach to describe the fluid flow at a macroscopic scale is developed. The principle is based on the representative volume element (REV), Bear (1972). At a microscopic level there are basically solid and non-solid (fluid or gas) components present (Section 2.1). The porosity at a solid-particle would be zero and one when located in a pore space respectively. When averaged over a larger volume, a volume that includes both solids and pore space, a value between these two extremes is obtained. As porous media are generally heterogeneous the REV is chosen so that the local heterogeneties in porosity are averaged. The sample volume (volume which is investigated), normally is much bigger than this REV (this can be several orders of magnitude). At a higher scale differences in heterogenous porous media (hydrogeological sections) become visible (see Figure 2.22).
Figure 2.22.: Porosity as function of different sizes of REV, ∆Ui at position i. (∆Uv )i is the local volume of fluid (from Bear, 1972). The scale of the samples used in this research are at meso scale, i.e. at the vertical dashed line in Figure 2.22, the field scale is at the outer right side of the graph where heterogeneity of the medium becomes noticeable.
22
2.4. NUCLEAR MAGNETIC RESONANCE (NMR) 2.4.3.2. Relation NMR and Porous Media
The NMR relaxation time of fluids in porous media are shorter than for bulk fluids (pure fluid). There is thus an influence of the pore wall on the NMR relaxation. This influence is caused by local magnetic fields close to the pore surface. Just as neighboring nuclei influence their local magnetic fields and cause the spins to relax (Section 2.4.1) the pore wall also has a local magnetic field. This magnetic field is about 1000 times stronger than the magnetic fields of the nuclei (Watson and Chang, 1997). Therefore the spins will relax much more quickly in the vicinity of the pore wall than in the bulk fluid (outside the influence range of the pore wall, which decreases to the power of six with distance Watson and Chang, 1997). The local magnetic field of the wall surface is mainly provided by paramagnetic materials in the grains that occupy sites (binding places) at the pore surface. The strength of this surface relaxation is the NMR surface relaxation parameter, ρ sur f ace [m/s]. Diffusion of the fluid molecules causes the spins to experience a change in magnetic field strength with place and time (Section 2.4.2). This results in a faster relaxation. The diffusion is dependent on the molecular diffusion constant, D0 (Section 2.4.2.5). Because of diffusion spins that have been relaxed at the pore wall are transported to the inner pore and unrelaxed spins are transported to the surface where they can relax. An overview of the different relaxation mechanisms in the NMR relaxation that play a role are in porous media are (Kenyon, 1997): • T sur f ace [s] , relaxation resulting from the pore wall (pore surface) contact (WR)
• T bulk [s], bulk relaxation (BR) is measured in a fluid when wall and gradient effects are not present. • T DR [s], relaxation shortened by molecular diffusion in an inhomogeneous (gradient) static magnetic field (DR). • T PR [s], relaxation is shortened by the presence of paramagnetic materials (PR) such as manganese and iron. Paramagnetic materials in the grain material are the main contributors to wall relaxation, decreasing T sur f ace ; paramagnetic molecules in fluids will shorten the bulk relaxation time, T bulk . Because the relaxation mechanisms work parallel to each other, T1 and T2 relaxation time constants can be given by (Bloembergen et al., 1947) 1 1 1 = bulk + sur f ace [1/s] T1 T1 T1
(2.34)
1 1 1 1 + [1/s] = + T2 T2sur f ace T2bulk T2DR
(2.35)
and
It has to be noted that the longitudinal relaxation time constant T1 is not affected by diffusion in a gradient field. The relation between relaxation and hydrogeological properties of water saturated sediments is given in Figure 2.23. The two key parameters are NMR surface relaxation, T sur f ace and surface-to-pore-volume (S/V, Kenyon, 1997). 2.4.4. Pore Size
For the relation between wall relaxation time and pore size it is assumed that the porous medium can be characterized with the following geometrical parameters: the volume and surface area of the pore, V and S respectively, with pore size a = VS ; the cross-Sectional area of a throat, A; the distance between centers of adjacent pores, q (See Figure 2.24, McCall et al., 1991).
23
2.4. NUCLEAR MAGNETIC RESONANCE (NMR)
NMR relaxation (T1,2) Surface relaxivity (rsurface )
Pore space properties (S,f )
- Permeability (k)
Surface-to-pore -volume ratio
- movable & irreducible water (Sw,irr) volumes
Figure 2.23.: Schematic overview of relation between the key parameters in NMR and hydrogeological properties (after Kenyon, 1997).
Figure 2.24.: Schematic of a porous-medium cross-Sectional area. The point at A forms a bottleneck for fluid flow (after McCall et al., 1991). As already discussed the behavior of NMR in porous media is controlled by three factors: the bulk relaxation time, T bulk , the surface relaxation time, T sur f ace , and the molecular diffusion relaxation time, T DR . If it is assumed that the molecular diffusion between pores (or interpore coupling rate) can be neglected (this is true if DR < WR), thus the average distance of a water molecule and the center of a nearby pore is large, and that the (Brownian) motion of the water molecules is fast, i.e. the average distance between the water molecules and the pore wall is small (fast diffusion limit), it can be proved that the relaxation in the pore can be written (Watson and Chang, 1997; Kenyon, 1997; Brownstein and Tarr, 1977): 1 Tsur f ace
=
1 Tbulk
+
ρsur f ace S [1/s] Vpore
(2.36)
Note that the bulk relaxation is considered either to be insignificant or can be removed by subtraction (equations 2.34 and 2.35) (Kenyon, 1997). When the parameter ρsur f ace (Section 2.4.3.2) is known (for either T1 and T2 ) and also the NMR relaxation time, pore size (p = V /S) can be calculated by equation 2.36. 2.4.4.1. Pore Size Distribution
In sediments there is not one single pore size (as can be seen in Figure 2.24), but a distribution of pore sizes (Kenyon, 1997). The wall relaxation Ms (t) can be written as the sum of the decaying signals for all pore sizes i
24
2.4. NUCLEAR MAGNETIC RESONANCE (NMR) 2500
α=0.001 α=0.01 α=0.1 α=1 α=10
Relative Amplitude
2000 1500 1000 500 0
2
2.5 3 3.5 4 4.5 Log Relaxation time constant [µs]
5
Figure 2.25.: Influence of regularization parameter (α ) on solution.
Ms (t) = ∑ Ai e
−
t 1,2 Ti
[A m]
(2.37)
i
Where: Ai [-] = number of protons in pores of size i (amplitude of signal) 1 ,T 2 = the corresponding decay time constant (T or T ) given by equation 2.36 T1,i 1 2 2,i
To obtain the so-called relaxation time distribution curve (or spectrum), A i versus Ti1,2 (equation 2.37) should be plotted. The problem of identifying Ti from equation 2.37 is a so-called ill-posed inverse problem: this means that the problem does not fullfill the definition of well-posedness as defined by Hadamard (1923): 1. the problem is solvable in the class of possible solutions 2. its solution is unique in this class 3. its solution is stable in this class with respect to admissible perturbations of the ingredients of the problem To (approximately) solve such an inverse and ill-posed problem a special theory is needed. This is known as Tikhonov regularization (Tikhonov and Arsenin, 1977). One important (or possibly the most important) part of this theory is how the regularization parameter, α (RP) is chosen. This should be done in such a way that the error in the data (in this case: Ms (t)) and in the operator (here: Ai ) is incorporated in the calculation of the RP according to the theorem of (Leonov and Yagola (1995) and thus the (approximate) solution. An example of the influence of the RP on the solution is given in Figure 2.25. Note that the higher the regularization term the more ”smoothed out” the solution gets. This is an example for did active purposes, i.e. the regularization parameter normally gets not higher than 1 and should be much smaller than 1 depending on the signal-to-noise ratio (S/N). The pore size distribution is thus a synonym for the relaxation time distribution curve when the fast-diffusion limit applies and each pore has its own relaxation time constant.
25
2.4. NUCLEAR MAGNETIC RESONANCE (NMR) 2.4.4.2. Porosity
From the NMR signal not only the decay times can be calculated but also the amplitude of the signal. The amplitudes must be compared to a reference sample. The reference sample is a strongly diluted copper sulphate solution in distilled water, a so-called ’doped sample’. The copper sulphate is paramagnetic and increases the relaxation of T2∗ ,T2 and T1 and shortens the total measurement time. The solution is diluted until a relaxation time of about 500 ms is obtained. It is important that the copper sulphate concentration is not too high, thus the relaxation time constant is not too short. For high concentrations the (normally) linear relation between ’amount of water molecules’ and ’amplitude of the NMR signal’ does not exist anymore and erroneous results are obtained. In S in Appendix D the relations between volume and amplitude for too short relaxation times and relaxation times that are long enough, are given. The weight of the water volume inside the reference sample is accurately known and used as reference for the other samples. From the amplitude of the sample, the amplitude of the reference and the water volume of the reference (VHre2 Of erence ), the water volume of the sample (VHsample )can be 2O calculated according to equation 2.38 (Straley et al., 1997): VHsample 2O
Asample VHre2 Of erence = Are f erence
(2.38)
When the pores of the sample are completely saturated then the porosity from NMR (φ NMR ) can be calculated with:
φ=
Vpore [−] Vbulk
(2.39)
2.4.4.3. Hydraulic Conductivity
The (intrinsic) hydraulic conductivity, κ can also be derived from the NMR relaxation times. For a detailed description of hydraulic conductivity see Chapter 2.2. The intrinsic hydraulic conductivity from NMR relaxation time has been estimated historically in three different ways 1. the estimation is based on the irreducible water volume, S wi (after Timur, 1968, 1969a,b):
κ=
φ4 2 Swi
(2.40)
2. the estimation is based on the porosity and the T2 relaxation time constant and it allows for fitting for different sediment types (adjustment of C-factor, after Kenyon (1997)):
κ = C φ 4 (T2 )2
(2.41)
3. An estimation based on porosity and T2 relaxation time constant (after Seevers, 1966):
κ = φ (T2 )2
(2.42)
Where: C = a factor dependent on sediment type The NMR relaxation time depends on the pore size as discussed in Sections 2.4.4 and 2.4.4.1. The adopted concept (Figure 2.23) does not include the S/V ratio. It would have been better that this factor was included because it determines the hydraulic conductivity, bottleneck principle, see figure 2.24). As porous media can be diverse and inhomogeneous some difficulties can be expected, i.e. similar
26
2.4. NUCLEAR MAGNETIC RESONANCE (NMR)
Single porosity system Water flow through the rock
1- 10 m Water flow through the rock
Double porosity system Figure 2.26.: Schematic overview of a single porosity and a double porosity model of permeability of aquifers (after Legchenko et al., 2002). NMR signals can lead to different permeabilities. This is illustrated in Figure 2.26, the connection between the pores of the double porosity models forms a bottleneck for groundwater flow and slows it down. These systems are at field scale level, but bottlenecks are present for single porosity media at microscopic scale also (Figure 2.24). Equations based on the length of the pore determined by NMR, L pore [m] are used to estimate the permeability (Kenyon, 1997): Vpore = ρsur f ace T1 [m] Lnmr = (2.43) S and this equation is squared to get the right dimensions
κ = C L2nmr = C(ρsur f ace T1 )2 [m2 ]
(2.44)
In this case the permeability is not only a function of T1 but also the surface relaxivity, ρsur f ace . In this expression (2.44), however, information about the surface-to-pore-volume ratio is not included. If this ratio is included as a factor it may lead to a better estimation of permeability (Sen et al., 1990): 1 2 L (2.45) F 2 nmr The factor F in this expression is the electrical formation factor (Section 2.3.2.1) defined by Archie (Archie, 1942) which carries information about the surface-to-pore-volume ratio. Note that equation 2.44 can only be applied when the surface relaxivity is constant, otherwise estimates of the permeability will vary (Kenyon, 1997).
κ =C
27
3. Material 3.1. Sample Material All measurements were performed on samples that can be divided in five different groups: 1. Synthetic samples. They exist of small glass pearls with three different diameters (0.01, 0.1 and 1.0 mm). Some of these are combined with clay. 2. Clayey samples. These samples are made from pure clay. 3. Artificially-made sand mixtures. Samples consisting out of different grain size ranges have been mixed (2-1, 1-0.5, 0.5-0.25, 0.25-0.125, 0.125-0.063, < 0.063 mm). 4. Samples taken from a bore core of a test site in Nauen Quarternary deposits from west of Berlin, Germany). 5. Coarse sands with a range of clay contents (3%, 5%, 10%, 15%, 20%). An overview of the groups and the samples they contain is given in Table 3.1 Table 3.1.: The five groups and the samples they contain. Group Group 1 Group 2 Group 3 Group 4 Group 5
Samples in group S01, S05, S1, S1-05, S1-DENMIX, S1-DENLAY, S1-MIX, S1-LAY CLAYROOM,CLAYDEN SANCON1-SANCON3, SANCOA-1 to SANCOA-5 NAUEN B1-15, NAUEN B1-22, NAUEN B1-23, NAUEN B1-27, NAUEN B1-28 3% CLAY, 5% CLAY, 10% CLAY, 15%CLAY, 20%CLAY
The first group has been taken as a control group; similar measurements have been conducted for glass pearls in a previous research by Krüger (2001). So, it forms a kind of control and check on reproducibility. The second group consits of clay. Clay has specific properties and in its pure form without sand these can be investigated. The third group of samples has been made in such a way that they are as close as possible to the Nauen samples (analyzed by Goldbeck (2002). This means that the grain spectra of the two samples are similar. The assumption is that in this way properties of Nauen samples can be explained by artificial samples. The samples taken in the fourth group are assumed to represent different hydrogeological regions in the subsurface of Nauen. Several geophysical methods have been applied to study the subsurface (Goldbeck, 2002). This information was used to pick samples from the boring (at depth of 15, 22, 23, 27 and 28 m depth below surface) that represent the different hydrogeological regions, i.e. aquifer, aquitard. As clay is a particular material with specific properties and pronounced influence on hydraulic conductivity, electrical conductivity and NMR parameters the fifth group has been made to investigate this. As explained above four different types of samples were used. The glass pearls are industry standard glass pearls. The glass pearls used are made by the company Sartorius. The glass pearls are cleaned with distilled water to remove the polish used during the production process. The polish has a strong effect on the electrical conductivity and therefore that must be cleaned. Coarse glass pearls
28
3.2. SAMPLE HOUSING
Figure 3.1.: A Fritsch analysette. Sieves with diameters of 0.063, 0.125, 0.25, 0.5, 1.0, 2.0 mm respectively. are cleaned quite fast (after three times of flushing with distilled water), for the small glass pearls this can take several weeks (Krüger, 2001). The quartz sand is bought at a construction market (Kluwe, Berlin, Germany) As all fractions are mixed, the raw material has to be sieved to obtain different grain size ranges. This is done with a sieve device (Fritsch analysette) with sieve diameters of 0.063, 0.125, 0.25, 0.5, 1.0, 2.0 mm respectively (see Figure 3.1). The coarse sand fractions already were sieved. For the finest fraction, however, some weeks of sieving had to be done for this study. The finer the grains the longer it takes for a certain volume. For two samples (Sancoa 4 and Sancoa 5) it would have taken too much time to obtain the volume needed. Therefore silt has been used from the Netherlands (southern part of the province Limburg). The Nauen samples were taken from a bore core. They have not undergone any special treatment or whatsoever. The clay used for the clay samples (group 2) and the “clay range” (group 5) was provided by the Faculty “Engineering Geology” of the Technical University of Berlin. It has been dried at 30 and 60 degrees Celsius (ClayRoom and ClayDen) respectively and pulvarised (crushed into small pieces) with a pulvariser (Fritsch pulverisette). No details about the origin of the clay or specific mineral content was available.
3.2. Sample Housing In this research several different measurements were performed. In order to compare and correlate these measurements to/with each other for all measurements the same sample has been used. The limiting size of the sample is formed by the Nuclear Magnetic Resonance (NMR) spectrometer used (Maran Ultra 2 MHz, Resonance Instruments UK). The diameter of the measuring bore is about 4.2 cm. The length of which the permanent magnet is (more or less) homogeneous is about 6 cm. Therefore the outer dimensions of the samples were chosen to be 6.0 x 4.0 cm (inner volume is 61 ml). The sample housing is made from makrolon tubes with 4 holes in it to make FDEC measurements possible. The samples can be closed with plastic caps and duct tape to prevent evaporation and leakage (see Figure 3.2).
29
3.3. SAMPLE PREPARATION Sample tube
4 Openings
6,0 [cm] 2,0 [cm]
1,0 [cm] 1,5 [cm] 0,5 [cm]
3,6 [cm] 4,0 [cm]
Figure 3.2.: Schematic drawing of the sample housing. Samples can be closed by a plastic cap (not shown in figure). 4 holes are for the FDEC measurements.
3.3. Sample Preparation The samples were filled with material, and weighted in a unsaturated (at about 100 degrees Celsius) oven-dry condition to obtain the dry weight. Then they were saturated. First this was done by evacuating all air out of the samples (with an exsiccator) and then flooding the samples. This method has two disadvantages. 1) during the saturation process material gets lost (samples were almost filled to the top), 2) the amount of water that entered the sample is unknown. Only after all the measurements were done the sample could be dried and the weight loss measured by a balance. One almost inevitable thing is that some material is lost during the SIP measurements, which introduces an error. To reduce the error the infiltration method was used: the water is injected by a needle until the sample is saturated. Excess of water at the top of the sample is sucked away by a syringe. In this way the amount of water added is known precisely and most of the samples have been saturated in this way. However, this method also has one disadvantage. The sample might seem to be saturated, but sometimes hidden air bubbles are trapped inside the sample. This problem can be overcome to a certain extent by gently tapping the sample on a hard surface, but this only works for coarse materials. A better way is to use an ultrasonic bath that ”cuts” the air bubbles into small pieces so that they can escape via the pores more easily. This method (with a Bandelin Sonorex rk 100) was used to saturate the samples measured by the permeamepermeaterter (see Section 4.2).
30
4. Measurement Methods In Figure 4.1 there is an overview of the relations between the measurements, the parameters and the hydraulic conductivity.
FDEC
ELECTRICAL RESISTIVITY + PHASE
SURFACE ANALYZER
PERMEAMETER
SURFACE-TO-PORE-VOLUME
PYCNOMETER
HYDRAULIC CONDUCTIVITY *
NMR
T2 T2 T1
WATER CONTENT/ POROSITY PORE SIZE DISTRIBUTION
Figure 4.1.: Relations between hydraulic conductivity and other measured parameters. A general remark: more (technical) information about the equipment itself can be found in their manuals.
4.1. Pore Space Properties The pore space measurements have been conducted to obtain an important ratio, the surface-to-porevolume (S/V) ratio. In Figure 4.1 the relation between the S/V is shown. 4.1.1. Density
The bulk density and the matrix density (Section 2.1.1) of the samples were not determined in the same way. The bulk volume of the sample has been determined by dividing the mass of the sample by the volume (57 ml of 61 ml was used). To check the degree of packing of the grains also the minimum and maximum bulk volume have been measured. The bulk volume was determined by measuring the volume of the grains by a graduated cylinder. As the bulk volume is strongly dependent on the packing (packing of the grains), the minimum and maximum bulk volume were measured by varying the degree of packing of the grains. To obtain the minimum bulk density the sample is gently ”poured” into the graduated cylinder to avoid a compaction. To obtain the maximum bulk density the graduated cylinder was tapped on a hard surface as long as no volume change was apparent anymore (Figure 4.2). The mass of the sample was measured by a balance. From the maximum and minimum bulk
31
4.1. PORE SPACE PROPERTIES
Maximum Volume
Minimum Volume Graduated Cylinder
Minimum Density
Maximum Density
Figure 4.2.: Relation between minimum and maximum volume and minimum and maximum density. volume the average bulk density is calculated and used in further calculations. The matrix bulk density for an unconsolidated sample is easy to obtain (although the technique behind is complicated). The mass can be accurately measured by a balance but the volume because of it’s irregular shape (as one grain is considered) cannot be easily calculated. Therefore a special device that is capable of measuring the volume of irregular shaped particles is used, a so-called pycnometer. This is a Gemini Accupyc 1330 (Figure 4.3).
Figure 4.3.: Gemini Accupyc 1330. A device that is able of accurately measuring the volume of irregular shaped particles (± 0.01 cm3 ). The Gemini Accupyc is a so-called ”gas displacement pycnometer”. Its principle is based on the expansion of (helium) gas in a calibrated control volume. A second volume contains the sample. Before the start of the measurement the pressure in both volumes is the same. Then an overpressure to the measuring volume is applied. Then a valve between the chambers is opened so the pressure falls to an intermediate value. The volume of the measuring volume can be calculated when the ”ideal gas law” is applied and rearranged in a suitable manner. For a more detailed description see Micromeritics (1997).
32
4.1. PORE SPACE PROPERTIES 4.1.2. Porosity and Water Content
The porosity (Section 2.1.2) of the samples can be calculated by the complementary relationship of equation 2.2. The volume of the grains (Vmatrix ) were known from measurements by the Gemini Accupyc 1330 and the bulk volume (Vbulk ) from the weight of the sample and the volume of the cylinder (see Section 2.1.1). The porosity of the samples have been obtained by dividing the difference in weight of sample in saturated state and oven-dry by the bulk volume of the sample (57 ml). From the maximum and minimum density (Section 2.1.1) the porosity a maximum and minimum porosity were calculated. The saturated water content was determined by the difference in mass of the dry and the saturated sample (1 gram water ≈1 ml). Some samples in the beginning of the experiment were saturated by an exsiccator. During the saturating some material from the samples was lost and therefore the mass of the added water could not be determined exactly. The saturated samples were weighted and when the measurements were finished the samples were dried and weighted again. The volume of the samples could easily be calculated as the sample housing is cylindrical with known dimensions (Figure 3.2). It was made sure that all samples had approximately the same volume (59 ml). The porosity was also determined by Nuclear Magnetic Resonance (see Section 2.4.4.2). 4.1.3. Specific Surface
The specific surface, S (2.1) is one of the most difficult properties of porous media to obtain. Specific surface is strongly dependent on the resolution of the measurement. It has the property that it increases with an increase in resolution. Structures that have this property are called fractal structures as introduced by Mandelbrot (Pape et al., 1987). The method used here is the BET method (after Brunauer et al., 1938). It has a resolution in the order of a nanometer as it is related to the cross-sectional area of a nitrogen molecule (10 −9 m2 ). The BET surface area was measured with a Micromeritics Gemini 2360 (see Figure 4.4).
Figure 4.4.: Micromeritics Gemini 2360 specific surface analyzer.
33
4.1. PORE SPACE PROPERTIES The principal of the method is that nitrogen gas when cooled down to its liquid state (by cooling the sample with liquid nitrogen) condensates/is adsorbed to the surface of the considered material. It will form a liquid nitrogen layer of only a few nitrogen molecules thick. Because of this condensation the nitrogen gas volume drops and therefore also the nitrogen pressure. The gas volume is determined with the same principle as used for the density measurement (Section 2.1.1) and the amount of nitrogen molecules adsorbed by the material can be calculated. This is explained in Brunauer et al. (1938). This measurement is quite time consuming as a careful preparation has to be done before the actual measurement can be performed. First, it should ensured that a reservoir with liquid nitrogen is available. Before the measurement can be performed the samples should be ”flushed” with nitrogen gas and evacuated with a vacuum pump to make sure that all air and water vapor have disappeared, i.e. the sample is dry. This has to be repeated several times and takes about 2 hours (half an hour evacuating the sample and 10 minutes flushing with nitrogen gas). These preparations were done with special designed equipment for this task, i.e. a Micromeritics Vacprep (see Figure 4.5).
Figure 4.5.: Micromeritics Vacprep. The evacuation is facilitated by a vacuum pump. After these preparations first a calibration measurement has to be executed to determine the partial nitrogen pressure of the air at air pressure of that moment (when the air pressure changes more then about 10 mbar during a measurement the calibration should be repeated). After the calibration the actual measurements can be performed. For coarse and smooth material (glass pearls, coarse sands), however, the specific surface is low and therefore the total surface is small. The device has a resolution limit for the total surface measured at about 0.1 m 2 and for the specific surface of about 0.01 m 2 /gr. When measured with the standard tube, the values for the coarse materials fell under the resolution limits. To overcome this problem a measuring volume with a larger volume then the standard volume was used. For this method the evacuation of the samples was done by a vacuum oven and the machine was partially rebuilt before the measurements could be performed. 4.1.4. Irreducible Water Saturation
Water in clayey samples is strongly bound by the clay minerals and can form so-called ”double-layer’. This water is not available for fluid flow and is called ”capillary-bound water”. When the pores of such a medium are drained or centrifuged to extract the water, only the part that is ”free”, i.e. not bound by the clay minerals, is retrieved. The part of the water that, given an under pressure, stays bounded to the clay mineral is the irreducible water content, S wi [-].
34
4.2. HYDRAULIC CONDUCTIVITY
4.2. Hydraulic Conductivity The hydraulic conductivity is determined by estimations from NMR (Section 2.4.4.3) and pore space properties (Hazen, Kozeny-Carman, Section 2.2.2). To verify these estimations a reference is provided by direct hydraulic conductivity measurements. 4.2.1. Falling Head Permeameter 4.2.1.1. Method and Equipment
The falling head permeameter (name has from permeability) is an experimental device to measure the hydraulic conductivity (Section 2.2) and is based on the falling head method. Figure 4.6 gives a schematic overview and Figure 4.8 a picture of the experimental device. Falling head indicates that the applied hydraulic head (see Section 2.2.1) is not constant over the measuring period, but it becomes smaller over time. This is due to the fact that the upper water level is dropping during the experiment (water flows from the burette through the sample and flows over in a graduated cylinder). To prevent that the sample falls into the tube connecting the sample, a filter is attached below the sample. This filter is half a centimeter thick and made of glass pearls with a diameter of 1 mm The type of filter that is chosen determines the maximum hydraulic conductivity that can be measured (k f ilter > ksample ). The falling head method is used for samples that are not so permeable (k < 2 m/d) . This applies to clayey sands, silts and clay samples. Samples that are more permeable should be measured with the constant head permeameter. The falling head permeameter is better suited to perform measurements that take a longer time period. Thus a smaller amount of water in the burette is required than for the constant head permeameter (mariotte bottle). It is not always easy to predict a priori how conductive a sample will be but with some experience or practice export knowledge is quickly developed. The glass pearl filter is connected to the bottom of the sample and sealed with duct tape. All air is let out of the system and the sample is saturated under a small hydraulic head difference. After the saturation the burette is refilled. When the outlet of the burette is opened the hydraulic head is applied and water flows over at the top of the sample. This water is captured with a graduated cylinder and a funnel. At the moment of opening the outlet of the buret the time is set at zero. As frequent and often as possible the time and corresponding water level (hydraulic head) is measured. The experiment ends when there is no hydraulic head difference anymore. 4.2.1.2. Post Processing Falling Head Data
The following equation is used to calculate the hydraulic conductivity from the falling head measurements: Aburette L H2 1 log (4.1) k= 86400 Asample (t2 − t1 ) H1 Where: t2 − t1 [s] = the time required for the head to drop from H1 to H2 L [m] = length of the soil column (sample length) Aburette [m2 ] = cross-sectional area of the burette Asample [m2 ] = cross-sectional area of the sample Note that the hydraulic conductivity should be calculated over the measuring range that the decrease in hydraulic head is linear on a plot where the time axis is logarithmic (see Figure 4.9).
35
4.2. HYDRAULIC CONDUCTIVITY Burette
Time t1 Water h1 Water
Time t2
h2
L
Sample
z1
H1
glass perl filter
z2
H2 valve /outlet
Reference Level (z=0)
Figure 4.6.: Falling head permeameter. Not shown here are a graduated cylinder and a funnel to capture the water after it flows over at the top of the sample. Patm
Cork
Bottle of Mariotte
Air reservoir Initial water level hdropping Final water level Hconst
Air bubbles
Water
L
glass perl filter
Sample zconst
valve /outlet
Reference Level (z=0)
Figure 4.7.: Constant head permeameter. Not shown here are a graduated cylinder and a funnel to capture the water after it flows over at the top of the sample.
36
4.2. HYDRAULIC CONDUCTIVITY
(a)
(b)
Figure 4.8.: Pictures of experimental permeameter device. All components are attached to a metal bar. a) the falling head permeameter, b) the constant head permeameter.
37
4.2. HYDRAULIC CONDUCTIVITY 0.81
Hydraulic Head [m]
0.805 0.8
H1
0.795 0.79 0.785 0.78
Dt
0.775 1
10
100
H2 1000
Time [s]
Figure 4.9.: Hydraulic head as function of time (logarithmic axis). 4.2.2. Constant Head Permeameter 4.2.2.1. Method and Equipment
Just like its ”sister”, the constant head permeameter is an experimental device to determine the hydraulic conductivity (see Figures 4.7 and 4.8 ). The constant head permeameter is better suited for more conductive samples (k > 2 m/d). This can be light clayey sands to permeable coarse sands. The upper limit of acceptable grain size is determined by the conductivity of the filter. One practical difficulty is how to obtain a constant head. One way to overcome this problem is to use an ”overflow system” . One reservoir is continuously filled and flows over so the water level in the reservoir is kept constant. There, however, is a more elegant way to obtain a constant head. The principle is called ”the bottle of Mariotte” (Maroto et al., 2002). The bottle is filled almost completely. Then the bottle (bottle of mariotte in Figure 4.7) is closed with a cork (air tight). In this cork a pipe is placed. In this pipe there is air pressure. When water is drained at the outlet of the bottle the water level in the bottle will drop. The air pressure in the air volume above the water level will drop also. The water level in the pipe will drop more quickly until the bottom of the pipe. Then small air bubbles come in and compensate the drop in air pressure caused by the drop in the water level. Although the water level is falling in the bottle, the pressure at the outlet of the bottle remains (almost) constant. This does not provide a truly constant hydraulic head but a pseudo-constant hydraulic head. The variations are caused by the size of the air bubbles coming in via the small pipe and are related to the pipe diameter. When the tube diameter is chosen sufficiently small also the variations in head pressure will be small. The fluctuations in head pressure are small and can be made even smaller when a ”damping reservoir” is connected in-between the Mariotte bottle and the sample. The height of the hydraulic head can be adjusted by varying the depth of the bottom of the pipe. Before the measurement the sample is saturated in the same way as for the falling head method. After saturation the mariotte bottle is refilled and closed again with the cork and pipe. When the outlet of the Mariotte bottle is opened there is a short period of non-constant head in the order of seconds. After this period the air bubbles are formed constantly with time. With a permeable sample a constant or steady-state situation is almost instantly achieved. The outflow is captured by a graduated cylinder and a funnel and for distinct volumes (i.e. every 100 ml outflow) the corresponding time is measured. When the water level in the Mariotte bottle has dropped till the bottom of the pipe the constant hydraulic head situation ends (and the system behaves as a falling head permeameter) and the measurement is stopped. 4.2.2.2. Post Processing Constant Head
To calculate the hydraulic conductivity Darcy’s law (equation 2.9) is slightly modified and rewritten:
38
4.2. HYDRAULIC CONDUCTIVITY
k=−
LQ ∆H
(4.2)
Where Q [m3 /d] is the discharge . Note that the hydraulic head gradient should be calculated when the system is in steady-state (approximately linear discharge (linear regression) see Figure 4.10). 700 600
Q=
Volume [ml]
500
DV Dt DV
400 300 200
Dt
100 0
200
400
600 800 Time [s]
1000
1200
1400
Figure 4.10.: Captured volume as function of time 4.2.3. PERO Permeameter
This permeameter (constant head approach), made by the PERO company, is able to measure the hydraulic conductivity automatically. It has been developed to measure samples with a low hydraulic conductivity (k < 2 m/d). This permeameter, in contradiction with the experimental device described before, is capable of applying a counter pressure on top of the sample which it allows to measure with a constant pressure. It has been adjusted by the workshop of Applied Geophysics, TU of Berlin, to measure smaller samples (dimensions specified in Section 3.2) . It consists of five functional units; a measuring cell that contains the sample, an analog manometer, an electronic pressure transducer and discharge meter housed in a measuring block, an electronically steered pressure installation and a data-logger connected to a PC (Figure 4.11).
39
4.2. HYDRAULIC CONDUCTIVITY
Figure 4.11.: PERO Permeameter. On the left (back) the analog manometer, on the right (back) the measuring cell, in the middle (front) the electronic measuring block (electronic pressure installation and data-logger are not visible in this picture, see also Appendix B.1) The PERO permeameter is controlled by the computer program ”Perolog 2000”. Before the preparations can be started first the sample should be fully saturated. This can be done with the experimental permeameter device at low hydraulic head difference (Section 4.2.1) or with the water injection method in combination with the ultrasonic bath (Section 3.3). After this the sample is mounted and the measuring cell filled with water. The tubes with over and under pressure are connected to the sample via connecting tubes at the outside of the measuring cell. An outside pressure that is higher than the over pressure (on the sample) is applied on to the water surrounding the sample. This is to ensure a good tightening of the sample. When all tubes are connected the pressures are applied according to the measuring scheme in the software. During this pressure building the sample and connecting tubes are ”flushed” to ensure all air is out of the system (see for a description of the ”building blocks” of the PERO permeameter Appendix B.1). When all this has been done the actual measurement can be performed by closing the valves of the electronic measuring block. All data is recorded by the data-logger or send to the PC directly. When the volume of the chamber in the measuring block has been emptied the measurement should be finished. The user, however, is not hinted when this happens but has to take a look on the values of the in- and outflow. When the in- and outflow are constant or declining the measurement should be finished and all data is written in an Microsoft Excel file. The
40
4.3. ELECTRICAL PROPERTIES data from the PERO permeameter are processed in the same way as for the constant head (Subsection 4.2.2.2).
4.3. Electrical Properties 4.3.1. Equipment
The equipment used to measure the frequency dependent electrical conductivity (FDEC, Section 2.3.1.2) is a Solartron SI 1260 (in short Solartron, Figure 4.12).
Figure 4.12.: Solartron SI 1260 with sample-holder to measure the frequency dependent electrical conductivity. Besides the measuring equipment a special sample holder was made. It has two brass electrodes at both sides of the sample and has 2 O-rings to make the sample watertight (sample holder visible in Figure 4.12). The electrodes that are used for the four point measurements (Telford et al., 1990) are made of a copper core/silver cover and connected with electric wires to the Solartron. For the two point measurements (Telford et al., 1990) the electric wires of the potential electrodes and the current electrodes are coupled and plugged in the two outsides of the sample holder and the Solartron respectively. For a schematic overview of the measurement setup see Figure 4.13.
41
4.4. NUCLEAR MAGNETIC RESONANCE (NMR)
Potential electrodes
Sample tube Config 1
Config 2
Config 3
Config 4
Film of scotch tape
Current electrodes (and for 2-point method also potential electrodes, config 5)
Figure 4.13.: Schematic overview of the measurement setup and the different configurations used for the frequency dependent electrical conductivity. The Solartron is operated by a PC which runs the software code Z60W. Several different measurements can be executed. The experiment type that is used is the ”sweep frequency with control voltage”. The frequency is ”swept” from 20 kHz to 0.1 Hz. To investigate the homogeneity of the sample (and simple to get better measurements) four different configurations were used for the 4-point measurement. Figure 4.13 shows the configurations. 4.3.2. Post Processing FDEC Data
Before the resistivity and phase were calculated from the measurements (equations 2.21 and 2.22) they first have to be corrected with a geometrical factor (just as the influence of potential electrode geometries on measurements (2-point, 4-point) in geo-electric field measurements, Telford et al. (1990)) depending on the potential electrode configuration. The geometrical factor can be calculated from the quotient of the surface of the current electrode and the distance between the two potential electrodes. Table 4.1 gives the geometrical factors. Table 4.1.: Geometrical factors used for FDEC measurements (see also figure 4.13). Configuration Config 1 Config 2 Config 3 Config 4 Config 5
Potential electrode distance [mm] 20 15 15 10 57
Geometrical factor [m] 0.0509 0.0679 0.0679 0.1018 0.0179
4.4. Nuclear Magnetic Resonance (NMR) If nuclear magnetic processes want to be observed an apparatus is needed that is able to emit RF pulses and receive the oscillating macroscopic magnetization vector. This device is known as NMR
42
4.4. NUCLEAR MAGNETIC RESONANCE (NMR) spectrometer. It contains a permanent magnet, transmit and receiver coils and electronics to operate it (for a detailed description about the hardware of a NMR spectrometer see Becker and Farrar, 1971). 4.4.1. Equipment and Operation
The NMR measurements in this thesis were conducted with a NMR Spectrometer from Resonance Instruments UK, a Maran Ultra 2 MHz (in short Maran, see Figure 4.14).
Bore
magnet box
heating
O.S.
Figure 4.14.: NMR Spectrometer from Resonance Instruments UK, a Maran Ultra 2 MHz. The device consists of three parts. In the left part the permanent magnet (magnetic field strength ≈ 1 0.047 T) is situated. The corresponding Larmor frequency (Ω H 0 ) of the H nuclei at this field strength is about 2 MHz. The permanent magnet and the transmitter and receiver coils (RF pulse generators) are visible in Figure 4.15. The part in the middle has a heating utility to keep the permanent magnets at a constant temperature (must be 5 degrees Celsius above room temperature, which is kept at 19 degrees Celsius. The right part contains the operating system (O.S.). 4.4.2. Calibration of Equipment
The software (RiNMR) that operates the NMR Spectrometer sends commands to the device and controls the data acquisition. The software is runned by a PC in a nearby room. The software has build-in macro’s and pulse sequences that can be selected and run automatically after start. Four of them were used to measure T2∗ , T2 , T1 and the molecular diffusion constant with respectively the pulse sequences FID, CPMG, INVREC and DIFF (Sections 2.4.2.1, 2.4.2.2, 2.4.2.3 and 2.4.2.5 respectively). Before any measurement can be conducted the Maran needs to be calibrated. This calibration is needed because the field strength of the magnet is not constant and therefore the pulse length of the RF pulses is not constant. Three macro’s are available to calibrate the Maran. In order to use the most accurate Larmor Frequency the deviation from the assumed magnetic field (0.047 T ) has to be measured 1 . The deviation from the 2 MHz value is automatically calculated and stored for later measurements. A calibration to obtain the right RF pulse length is used to rotate the spin polarization by 90 degrees or π /2 radians and 180 degrees or π radians 2 . The pulse length that corresponds to the maximum amplitude for the 90 or 180 degree pulse is used and automatically stored for later measurements. For more 1 This 2 The
is done with the macro ’.AUTOO1’ macro ’.AUTOP90’ measures the amplitude of the NMR signal for different pulse lengths.
43
4.4. NUCLEAR MAGNETIC RESONANCE (NMR)
il Co
Ma
il Co
et gn Figure 4.15.: View inside the Maran Ultra 2 MHz. In the middle the measuring bore (white hole) with at both sides the permanent magnets and the send/receiver coils (red color). details about measurement settings see Krüger (2001) or the users manual, Resonance Intruments (2000). For an overview of the pulse sequences see Section D.2 in Appendix D. 4.4.3. Post Processing NMR Data
To process the data obtained by the acquisition program RiNMR (Section 4.4.2) three software package are available. Winfit is the program used to calculate the relaxation time constant and the amplitude of the NMR signal. WinDXP is used to calculate the relaxation time distribution (pore size distribution, Section 2.4.4.1). RiDiff is used to calculate the molecular diffusion constant, D 0 (Section 2.4.2.5). 4.4.3.1. RI Winfit for Relaxation time constant
The functions used by the software RiWinfit to obtain T2∗ , T2 and T1 (Section 2.4.2) is given in equations 4.3, and 4.4 respectively:
y = Σ i Ai e
−
y = Σ i Ai 1 − e
44
x T2,n
−
x T1,n
(4.3) !
(4.4)
4.4. NUCLEAR MAGNETIC RESONANCE (NMR) Where: y [-] = data Ai [-] = Amplitude at time i T1,2,i [s] = relaxation time constants at time i n = number of exponentials The program can display the recorded data. An important processing step, i.e. data quality control, is done first by analyzing the NMR signal. For instance, was the S/N ratio high enough, and has the complete decay curve been recorded? Often this program is used to have a quick check on the parameters with a low amount of stacks (repeated measurements to increase signal-to-noise ratio (S/N), referred to as Number of Scans by RiNMR) before the ’true’ measurement was started. When the measurement was approved it was used to calculate the Amplitude of the signal and the relaxation time constant: the decay of the signal is approximately exponential and hence an exponential curve is fitted to the data. The program allows for as much as four exponential terms to be fitted. A visual check on the quality of the fit is given as well as the standard deviation and sum of the residuals. When a one-exponential fit is not good enough the user can try a multi-exponential fit. The initial amplitude is a measure of the total amount of protons present in the sample. 4.4.3.2. RI WinDXP for Relaxation Time Distribution
This program is used to solve the inverse ill-posed problem of relaxation time constant distribution or pore-size distribution as explained in Section 2.4.4.1. That is to find the amplitudes Ai at relaxation time constants Ti (equation 2.37). The algorithm used to calculate A i is to minimize the sum of the squared residuals is: 2 ΣN i=1 wi
Z
D
k(Ai ,t) f (t)dt − gi
2
= min
2
N f (t)2 + α Σt=1
(4.5)
Where w2i the weights, are inversely proportional to the (known) measurement error variances. If the noise variances are equal it results in equal weights, w i = 1 for all i. To prevent that the noise in the data leads to high frequency (non-physical) artefacts a regularization term to equation 4.5 is added to stabilize the solution (Tikhonov and Arsenin, 1977).
min =
2 ΣN i=1 wi
Z
D
k(Ai ,t) f (t)dt − gi
(4.6)
Where the second term of equation 4.6 is the regularization term and α is the regularization parameter, α (Tikhonov and Arsenin (1977)). The regularization parameter can be seen as a weighing parameter. It is a trade off between the influence of the data at one side and the noise of the data on the other side, and when α is too small, the stabilizing effect of the regularization parameter is lost and the solution will be overly sensitive to noise in the data; when α is too large the solution is dominated by the regularization parameter and will be artificially smooth (Figure 2.25). How the regularization parameter is calculated by winDXP and further details see Butler et al. (1981). Note that a regularization parameter higher than 1 indicates noisy data. The regularization theory is only applicable in cases with a low noise level and therefore the (regularized) solution should be treated really carefully.
45
4.4. NUCLEAR MAGNETIC RESONANCE (NMR) 4.4.3.3. RiDiff for Molecule Diffusion Constant
To calculate the diffusion constant several experiments are performed with different pulse gradient lengths (Section 2.4.2.5). The amplitude of all these measurements is calculated by the software program RiDIFF and plotted in a graph with (Resonance Intruments, 1999): • x-axis: (γ proton g)2 δ 2 (∆ − δ /3) • y-axis: ln (echo amplitude) The diffusion is the gradient of the best linear fit to this graph. This diffusion coefficient is automatically calculated by the software RiDiff. It should be noted that the temperature of the sample that is measured should be stabilized before the measurement is performed. The diffusion coefficient is strongly dependent on the temperature as it effects the Brownian motion. It should be warmed or cooled to a temperature of 25 degrees Celsius as that is the temperature in the measuring bore of the Maran (with an in-build heating). Better diffusion measurements for different temperatures could have been made if a special cooling and heating element could have been used. At the moment, however, this element is not (yet) available at the Department of Applied Geophysics, TU of Berlin. Another requirement to perform diffusion measurements is the presence of a magnetic field gradient. Normally a linear gradient field is chosen. For the Maran a Crown Macrotech magnetic field gradient amplifier is used. This amplifier is automatically controlled by the operating system of the Maran. Magnetic field gradients in only one direction are possible. To allow for 2- or 3-dimensional magnetic field gradients (for magnetic resonance imaging (MRI) experiments) one and two extra amplifiers are needed, respectively . For specific details about measuring the diffusion coefficient see Resonance Intruments (1999).
46
5. Results and Discussion The five different groups used in this research each have their own symbols in the figures. The same symbols for NMR T2∗ , T2 and T1 decay times and FDEC 2-point and 4-point measurements are used but it should be clear from the context if it concerns NMR of FDEC measurements. In some cases the system cannot be used, then the rule that when the legend is within the graph itself another (local) system is used. When the legend is located outside the graph, the global system applies. However, it is tried to be as consistent as possible and whenever possible the global system has been used. See Table 5.1 for an overview of the groups and their corresponding symbols and Table 3.1 for the samples they contain. Table 5.1.: Explanation of the symbols used in the figures in this thesis for the identification of the groups. Their synonyms are also listed.
! " # %
$ %!
& #
%
5.1. Relation Specific Surface and Clay Content In Figure 5.1 the relation between the specific surface (S m , Section 2.1.3) and clay for the samples in group 5 is shown. The relation is almost one on one. The clay dominates the specific surface, not only for coarse sands but for all samples (Table A.2 in Appendix A).
47
5.2. RELATION HYDRAULIC CONDUCTIVITY AND RESISTIVITY 18
Measured f(x)=0.8967*x, R2=0.9995
16
Sm [1/µm]
14 12 10 8 6 4 2 0
0
5
10
15
20
Clay [%]
Figure 5.1.: Specific surface (Sm ) as function of clay content of a coarse sand sample (d 1.0-0.5 mm).
5.2. Relation Hydraulic Conductivity and Resistivity In Figure 5.2 the relation between resistivity (ρ res , Section 2.3.1.1) and specific surface (S por ) is plotted. 10000
Resistivity [ Wm]
2 1000
1
100
Synth Synth Clay Clay Sand mix Sand mix Nauen Nauen -0.5434 148.4*x
10
1 0.001
0.01
0.1 1 Spor [1/µm]
10
100
Figure 5.2.: Resistivity as function of surface-to-pore-volume ratio, S por . As interface conductivity (σinter f ace , Section 2.3.2.1) is related with the surface of the grains, the resistivity decreases with an increase of specific surface. Two clusters can be identified: a cluster with a resistivity in the range 20-40 Ωm with the specific surface from about 1-60 1/µ m and a cluster with specific surface around 0.8 1/µ m and a corresponding resistivity in the range 200-3000 Ωm. The first cluster shows an independent resistivity behavior, the second cluster a high variation for intermediate specific surfaces. For the sand mixtures the information of the S/V ratio does not seem to be incorporated in the resistivity. The Nauen samples do show a relation between S/V ratio and resistivity. In Figure 5.3 and Figure 5.4 the measured hydraulic conductivity (Section 4.2) and the hydraulic conductivity calculated from NMR (Sub-section 2.4.4.3) as a function of resistivity are plotted.
48
5.2. RELATION HYDRAULIC CONDUCTIVITY AND RESISTIVITY
k-Measured [m/d]
1000 100
1
2
10
Synth Clay Sand mix Nauen 1.0859 0.0296*x
1 0.1 0.01 1
10
100
1000
10000
Resistivity [W m]
Figure 5.3.: Measured Hydraulic conductivity as function of Resistivity, ρ . 1000
k-NMR [m/d]
100
1 10
2
1
Synth Synth Clay Clay Sand mix Sand mix Nauen Nauen 1.256 0.0216*x
0.1 0.01 1
10
100
1000
10000
Resistivity [W m]
Figure 5.4.: Hydraulic conductivity from NMR as function of Resistivity, ρ . Both show a positive correlation with the resistivity. Clay has the lowest resistivity of all samples, the resistivity of the other samples increases with a decrease in clay content (Appendix A.1). Clay not only has a strong influence on the resistivity but also on hydraulic conductivity (Appendix A.3). Just as for the relation between resistivity and specific surface (S m ), here also two clusters can be identified: Resistivity range 20-40 Ωm and a hydraulic conductivity around 100 m/d. The samples in the second cluster correspond with the samples of the first cluster the first cluster in Figure 5.2 and the samples of the first cluster with the samples of the second cluster in Figure 5.2. A constant S/V ratio results in a constant hydraulic conductivity and a varying S/V ratio in a varying hydraulic conductivity. Both for Figure 5.3 and Figure 5.4. This can be explained with the S/V ratio as it forms the bottleneck for hydraulic conductivity (Figure 2.24). In Figure 5.5 the resistivity as function of specific surface for varying fluid conductivities (σ f luid ) at 100 Hz is plotted.
49
5.2. RELATION HYDRAULIC CONDUCTIVITY AND RESISTIVITY
Resistivity [Ωm]
1000
0.0038 [mS/cm] 1 [ms/cm] 10 [ms/cm] 100 [ms/cm]
100 10 1 0.1
2
4
6
8
10 12 Sm [1/µm]
14
16
18
Figure 5.5.: Resistivity at 100 Hz (4-point measurements) as function of specific surface for clay range samples (group 5). For relative low conductive fluids (0.0038 and 1 mS/cm, respectively) the resistivity decreases with an increase in specific surface, i.e. the clay content (σ inter f ace , Section 2.3.2.1) is the main conductor. For higher fluid conductivities this is not the case and the resistivity is about constant. Exceptions are the resistivity of the 3% clay sample. Two-point measurements at 100 Hz show similar behavior though more biased. Similar characteristics are observed for both 2- and 4-point measurements at 10000, 1000 and 10 Hz (see figures C.1a and C.1b in Appendix C, respectively). This means that there is almost no frequency dependence for these measurements although a slight increase with resistivity for a decrease in frequency is observed. In Figure 5.6 the relation between resistivity (4point measurement) as function of fluid conductivity for varying clay percentages is shown.
50
5.2. RELATION HYDRAULIC CONDUCTIVITY AND RESISTIVITY
Resistivity [Ωm]
10000
3 % Clay 5 % Clay 10 % Clay 15 % Clay 20 % Clay
1000 100 10 1 0.001
0.01
0.1 1 Conductivity [mS/cm]
10
100
Figure 5.6.: Resistivity at 100 Hz (4-point measurement) as function of fluid conductivity, σ f luid for Clay range samples (group 5). The lower the clay content the more pronounced the drop in resistivity with increasing fluid conductivity, except for the samples with a clay content of 15% and 20% at a conductivity of 100 mS/cm that are interchanged. Strange enough this is not true for 2-point resistivity measurements that show almost an opposite behavior (Figure 5.7).
Resistivity [Ωm]
10000
3 % Clay 5 % Clay 10 % Clay 15 % Clay 20 % Clay
1000 100 10 1 0.001
0.01
0.1 1 Conductivity [mS/cm]
10
100
Figure 5.7.: Resistivity at 100 Hz (2-point measurement) as function of fluid conductivity, σ f luid for Clay range samples. The resistivity behavior of the 4-point measurements seems to be more likely because the lower the clay content the higher the influence in conductivity, i.e. samples with a higher clay content already
51
5.2. RELATION HYDRAULIC CONDUCTIVITY AND RESISTIVITY have a better conductance. This statement is not supported by the behavior of the 2-point resistivity measurements. For the first increase in the conductivity (Figure 5.5) from 0.0038-1 mS/cm the decrease is only half an order of magnitude while for further increase of the fluid conductivity this is an order of magnitude. The 2-point resistivity measurements (Figure 5.6) do not show a clear behavior. In Figure 5.8 and b the resistivity and phase for 4-point measurements are plotted as function of frequency. 160 140
3% Clay 20% Clay
15
120 Phase [ o ]
Resistivity [Ωm]
20
3% Clay 20% Clay
100 80
10 5
60 0
40 20
0.1
1
10 100 Frequency [Hz]
1000
-5
10000
(a) Resistivity as function of frequency
0.1
1
10 100 Frequency [Hz]
1000
10000
(b) Phase as function of frequency
Figure 5.8.: Resistivity and phase as function of frequency for 4-point measurements with 0.0038 mS/cm fluid conductivity. The standard deviation (and this applies also for Figure 5.10 ) has been calculated from the four different configurations (see Section 4.3.1). It gives as such an indication of the homogeneity of the sample. The measurement error of the Solartron is unknown but will be mostly smaller than the standard deviation of the four different configurations and/or coupling resistivity. The resistivity of the sample with a clay content of 20% shows a slight increase of the resistivity with a decrease in frequency. For low frequencies a ”constant” current leads to the hydrolyse of the sample fluid, i.e. the sample is short-cut which results in an increase of resistivity. For the sample with a clay content of 3% this effect is however not absorbed but this measurement is involved with a large standard deviation. Note that the standard deviation for the 20% clay sample is much lower than for the 3% clay sample, for both resistivity as phase. The 20% clay sample is also much more ”stable”. In Figure 5.9 the resistivity and phase as function of frequency for 2-point measurements is plotted.
52
5.2. RELATION HYDRAULIC CONDUCTIVITY AND RESISTIVITY
10000
0
3% Clay 20% Clay
3% Clay 20% Clay
1000
Phase [ o ]
Resistivity [Ωm]
-5
100
-10 -15 -20
10
0.1
1
10 100 1000 Frequency [Hz]
-25
10000
0.1
(a) Resistivity as function of frequency
1
10 100 Frequency [Hz]
1000
10000
(b) Phase as function of frequency
Figure 5.9.: Resistivity and phase as function of frequency for 2-point measurements with 0.0038 mS/cm fluid conductivity. As for the 2-point measurements only one possible configuration is possible it is not possible to calculate a standard deviation. The resistivity increases with a decrease in frequency for both the 3% and 20% clay sample; the ”hydrolyse” effect. The phase for the 3% and 20% clay content varies strongly (normally a phase effect of a few degrees is observed) and strange enough are opposite to each other. It is assumed that these measurements are strongly biased and have nothing to do with a”true” frequency effect. In Figure 5.10 the resistivity and phase for 4-point measurements as function of frequency is plotted. 2 1.5
3% Clay 20% Clay
15 10
1
Phase [ o ]
Resistivity [Ωm]
20
3% Clay 20% Clay
0.5 0
5 0 -5 -10
-0.5 -1
-15 0.1
1
10 100 Frequency [Hz]
1000
-20
10000
(a) Resistivity as function of frequency
0.1
1
10 100 Frequency [Hz]
1000
10000
(b) Phase as function of frequency
Figure 5.10.: Resistivity and phase as function of frequency for 4-point measurements with 100 mS/cm fluid conductivity. The observed resistivity behavior is opposite as the resistivity in Figure 5.8. The standard deviation of the 20% clay sample is again much lower than for the 3% clay sample. Note that a negative resistivity is physically impossible but can be calculated statistically (error bars have a ρ < 0). The resistivity for the 20% clay sample is two times as high as the sample with only 3% of clay, it would
53
5.3. HYDRAULIC CONDUCTIVITY AND NMR PROPERTIES OF SAMPLES be expected to be interchanged. The corresponding phase is more ”stable” than for the lower fluid conductivity and more alike for the 3% and 20% clay samples. In Figure 5.11 the resistivity and phase for 2-point measurements as function of frequency is shown. 100
0
3% Clay 20% Clay
Phase [ o ]
Resistivity [Ωm]
-5
10
-10 -15 -20
1
0.1
1
10 100 Frequency [Hz]
1000
-25
10000
(a) Resistivity as function of frequency
3% Clay 20% Clay 0.1
1
10 100 Frequency [Hz]
1000
10000
(b) Phase as function of frequency
Figure 5.11.: Resistivity and phase as function of frequency for 2-point measurements with 100 mS/cm fluid conductivity. The Resistivity increases with a decrease in frequency as observed before. The phase decreases with a decrease in frequency. This is consistent for both the 3% and 20% clay sample though this is only apparent for the 2-point measurements and is not found for 4-point measurements. The resistivity for the 3% and 20% clay sample for the2-point measurement is 2-7 and 20 times higher than the 4-point measurements respectively. For the 2-point measurements a consistent higher resistivity is measured for both the low conductive as high conductive than for the 4-point measurements. This difference is a result of the higher coupling resistivity for the 2-point measurements.
5.3. Hydraulic Conductivity and NMR properties of Samples In figures 5.12, 5.13, 5.14 the porosities of the groups ”synthetic samples”, ”sand mixtures” and ”Nauen samples” obtained by NMR (Equation 2.39) are plotted as function of the porosity obtained by weighing the samples (Section 4.1.2).
54
5.3. HYDRAULIC CONDUCTIVITY AND NMR PROPERTIES OF SAMPLES 0.6
Porosity by NMR [-]
0.55 0.5 0.45
T2* T2 T1 Y=X
0.4 0.35 0.3 0.25 0.2 0.15 0.1
0.15
0.2 0.25 Porosity by weighing [-]
0.3
Figure 5.12.: Porosity of Synthetic mixtures (group 1) and clay (group 2) obtained by NMR and weighing.
Porosity by NMR [-]
1 0.8
T2* T2 T1 Y=X
0.6 0.4 0.2 0 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 Porosity by weighing [-] (a) Sand mixtures
Figure 5.13.: Porosity of Sand mixtures (group 3) obtained by NMR and weighing.
55
5.3. HYDRAULIC CONDUCTIVITY AND NMR PROPERTIES OF SAMPLES
0.6
T2* T2 T1 y=X
Porosity by NMR [-]
0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1
0
0.05
0.1 0.15 0.2 0.25 Porosity by weighing [-]
0.3
0.35
(a) Nauen samples
Figure 5.14.: Porosity of Nauen samples (group 4) obtained by NMR and weighing. The porosities obtained from NMR agree reasonably well to fairly well (T2 average error of ≈10%, T1 ≈ 15%) for most of the samples but the porosities obtained from T2∗ deviates (T2∗ average error of ≈ 60%). The NMR porosities were calculated by fitting the NMR relaxation time (Section 4.4.3.1) . The steepness of this relaxation decay depends on the relaxation time constant, the shorter the steeper. When fitting the decay curve the root mean square error of the initial amplitude (Section 4.4.3.1) in the fit will be highest for fast relaxing samples, i.e. a high change in amplitude for a small change in the fit. The relaxation times of the groups become smaller from synthetic samples to sand mixtures to Nauen samples (Appendix D.3). This supports the ”error of fit” explanation. An overview of the porosities of all samples together is plotted in Figure D.12 in Appendix D. The hydraulic conductivity from NMR for the clay range is plotted in Figure 5.15. 100
Clay Range
K-T2 [m/d]
10
1
0.1
0.01
0
10
20
30 40 Sm [1/µm]
50
60
Figure 5.15.: Hydraulic conductivity calculated from NMR-T2 measurements for the clay range and specific surface, Sm . The hydraulic conductivity shows an exponential decay with the specific surface (S m ) or an increase
56
5.3. HYDRAULIC CONDUCTIVITY AND NMR PROPERTIES OF SAMPLES in clay content (the relation between T2 and clay content is similar, see Figure D.13 in Appendix D). The drop in hydraulic conductivity for the samples with 3%, 5% and 10% clay is one order of magnitude and the corresponding difference in specific surface is almost 4.5 1/µ m. For the samples with a clay content of 10 %, 15%, 20% and 100% (pure clay sample also included) the hydraulic conductivity is almost constant with the difference in specific surface of about 50 1/µ m, ten times as much as for the light clayey samples. The hydraulic conductivity does not change much from about 10% of clay or a specific surface of 8.5 1/µ m. This exponential decay of hydraulic conductivity is also found for similar samples by Slater and Lesmes (2002). The NMR decay time, T2 (Section 2.4.2) dependence on specific surface is plotted in Figure 5.16.
NMR Decay Time [ms]
10000
Synth Synth Clay Clay Sand mix Sand mix Nauen Nauen 159.6*x-0.6755
1000 100 10 1 0.001
0.01
0.1 1 SPor [1/µm]
10
100
Figure 5.16.: NMR decay time as function of surface-to-pore-volume, S por . Most of the samples are within half an order of magnitude of the fitted function. The samples with silt and a high content of grains smaller than 0.063 mm are situated below the fitted line. Samples that are more coarse are above the line, with the exception of the glass pearls. The specific surface of the glass pearls is ”too small” for their relaxation time. The deviation from the more natural samples is considerably (similar behavior is shown between hydraulic conductivity from NMR and specific surface, see Figure D.14). The strong deviation from the fitted line has two reasons: 1) the glass pearls have a smooth surface, thus a low specific surface, lower than natural samples which have ”micro scratches” that increase the specific surface. 2) because of this low specific surface the surface relaxation, T sur f ace (Section 2.4.3.2), is low and the spins can exist longer. The hydraulic conductivity calculated from NMR (Section 2.4.4.3) shows a good agreement with the measured hydraulic conductivity for the clay samples and the glass pearls (Figure 5.17).
57
5.3. HYDRAULIC CONDUCTIVITY AND NMR PROPERTIES OF SAMPLES
K Calculated [m/d]
1000
Synth Synth Clay Clay Sand mix Sand mix Nauen Nauen Y=X
100 10 1 0.1 0.01 0.01
0.1
1 10 K Measured [m/d]
100
1000
Figure 5.17.: Hydraulic conductivity calculated from NMR as function of measured hydraulic conductivity obtained with permeameter. There is a scatter of samples of intermediate hydraulic conductivity. This scatter consists out of Nauen samples and the sand mixture’s (”Nauen like”). The silty samples (Sancoa-4 and -5,see Appendix A.1) have a ”too low” relaxation time for their hydraulic conductivity. Their relaxation time is about three times that of clay but their hydraulic conductivity is an order higher on the average. The Nauen samples have a too high relaxation time or a too low hydraulic conductivity, but the first assumption is more likely as the relaxation time is also higher than expected for their specific surface (see Figure 5.16). Because silt is a fine fraction ( 0.002 < d < 0.063 mm) their pore size is also small, this results in a short relaxation time. Unlike clay, silt does not have a ”double-layer” that makes the pores very narrow and slows down the hydraulic conductivity. This is probably the reason for their difference in hydraulic conductivity. The coarse sand samples (Sancoa-1, -2 , -3 and Nauen B1-15, B1-22 and B1-27) show a ”too high” calculated hydraulic conductivity. It seems that the large pores determine the mean relaxation time but for the ”true” hydraulic conductivity the poor sorting of the sample is important (poor sorting decreases the hydraulic conductivity).Straley et al. (1997) found a pre-factor (C, in equation 2.44) of about 4.6 for sandstones. In this research a factor of 1 was applied that provide a good fit for clay and coarse materials. Another possibility to explain the scatter in hydraulic conductivity is the variation of surface relaxivity between the samples. The glass pearls, as already discussed, have a low specific surface that explains the long relaxation time. The natural samples consist of material of different origin; clean sands with a low amount of paramagnetic material (group 3) and sands that might be covered with a paramagnetic ”skin” (group 4), because of its strong magnetic influence only a small amount of paramagnetic material is enough to have a strong influence on the relaxation time (Section 2.4.3.2). The calculated hydraulic conductivity after Kozeny-Carman (equation 2.14) and Hazen (equation 2.15) models are maximally 3 orders of magnitude apart for low hydraulic conductivities and almost one for the high hydraulic conductivities (Figure 5.18, similar relation for T1 , see D.15 in Appendix D).
58
5.3. HYDRAULIC CONDUCTIVITY AND NMR PROPERTIES OF SAMPLES 10000
k-calculated [m/d]
1000 100 10 1 0.1 0.01 Kozeny-Carman Hazen Hazen fitted Koz-Car fitted
0.001 0.0001 1e-05
1
10
100 T2 [ms]
1000
10000
Figure 5.18.: Hydraulic conductivity estimated by Kozeny-Carman and Hazen as function of NMR-T2 . When Figure 5.18 is estimated with a plot of hydraulic conductivity estimated and measured as function of specific surface it becomes clear which of the two models is best (Figure 5.19).
Hydraulic Conductivity [m/d]
10000 1000 100 10 1 0.1 0.01 0.001 0.0001 1e-05 0.001
Koz-Car Hazen Measured Hazen fitted Koz-car fitted Measured fitted 0.01
0.1 1 Spor [1/µm]
10
100
Figure 5.19.: Hydraulic conductivity (measured and estimated) as function of surface-to-porevolume, S por . The hydraulic conductivity estimated by Hazen is much closer by the ”true” hydraulic conductivity than the more sophisticated Kozeny-Carman model. The estimation of both models is better for higher hydraulic conductivities (sand instead of clay or silt). Both models are developed to estimate the hydraulic conductivity of sandy samples. So, the result is not surprising. Similar results of the Kozeny-Carman and Hazen models are obtained by (Slater and Lesmes, 2002). They also found that a Hazen type equation provides order of magnitude hydraulic conductivity estimates compared with measured hydraulic conductivity and that for poorly sorted materials the estimation is the poorest. The measured hydraulic conductivity shows a negative relationship with the Surface-to-pore-volume, S por (Figure 5.20).
59
5.3. HYDRAULIC CONDUCTIVITY AND NMR PROPERTIES OF SAMPLES
k-Measured [m/d]
1000 Synth Clay Sand mix Nauen -0.8953 4.211*x
100 10 1 0.1 0.01 0.001
0.01
0.1 1 Spor [1/µm]
10
100
Figure 5.20.: Measured Hydraulic conductivity as function of surface-to-pore-volume, S por . Samples with a hydraulic conductivity between 0.7-3 m/d are more or less independent from the specific surface. This is comparable with the behavior observed for clay (see Figure 5.15). Note that the drop in hydraulic conductivity for the clay range is considerably in the range 0.2-5 1/µ m while here the samples in the middle range are more or less constant over that specific surface range. This is probably because the relative influence of the clay on the relaxation time is much higher for coarse samples than for samples with a considerable amount of fine grains. The mechanisms of both different sample types might be completely different. Clay seems to reduce hydraulic conductivity because of increased specific surface, while the fine particles seem to have an effect on the mean pore size. For an overview of the NMR decay times of all samples see figures D.10 and D.11 in Appendix D. An overview of the NMR relaxation time curves Section D.3 is given in Appendix D. In Figure 5.21 the pore size (relaxation time distribution, Section 2.4.4.1) for a layered sample (glass pearls on top of clay) is shown.
Relative Amplitude
2500 2000
Clay
1500 Glass perls
1000 500 0 1
10
100
1000
10000
Time [ms] Figure 5.21.: Pore size distribution of layered sample, Denlay (glass pearls on top of clay). A separate peak is resolved for clay and glass pearls; they behave independently. When the same
60
5.3. HYDRAULIC CONDUCTIVITY AND NMR PROPERTIES OF SAMPLES material is mixed however only the peak of clay is observed with a mean relaxation time constant equal to that of pure clay (see Figure 5.22).
Relative Amplitude
6000 5000 4000 3000 2000 1000 0
1
10
100 Time [ms]
1000
10000
Figure 5.22.: Pore size distribution of sample (S1-Denmix) with clay and glass pearls mixed. The cross in the figure shows the mean relaxation time (as calculated by RiWinfit, Section 4.4.3.1) The glass pearls are ”masked” in this way and unobservable by NMR alone, so, here the combination with FDEC measurements becomes interesting. In figures 5.23a and b the electrical properties for clay with and clay without glass pearls are shown respectively.
2
10
0
-2
5
Phase [°]
15
Resistivity 4-point Resistivity 2-point σ’ 4-point σ’ 2-point σ’’ 4-point σ’’ 2-point Phase [°] 4-point Phase [°] 2-point
4 15
4
2
10
0
-2
5
-4 0
0.1
1
10
100
1000
Phase [°]
20
Resistivity 4-point Resistivity 2-point σ’ 4-point σ’ 2-point σ’’ 4-point σ’’ 2-point Phase [°] 4-point Phase [°] 2-point
Log Resistivity [Ωm]
Log Resistivity [Ωm]
20
-4 0
10000
Frequency [Hz]
0.1
1
10
100
1000
10000
Frequency [Hz]
(a) Electric properties for clay sample with glass perls (S1-Denmix)
(b) Electric properties for clay sample without glass perls (Denclay)
Figure 5.23.: Electrical properties for clay with and clay without glass pearls respectively. However, the resistivity and the phase for clay with, and clay without glass pearls shows virtually no difference; the clay also dominates the NMR relaxation time as well as the electrical resistivity.
61
5.3. HYDRAULIC CONDUCTIVITY AND NMR PROPERTIES OF SAMPLES For SNMR measurements this phenomenon is important as although the same signal is received the content of the subsurface might be quite different. For the sand samples the U-factor has been calculated (Section 2.2.2) and classified in a sand fraction (coarse sand, medium coarse sand, medium fine sand, etc.). Together with the clay and silt samples the relaxation times observed by NMR (this thesis) and from SNMR as obtained by Schirov et al. (1991) are listed in Table 5.2. Table 5.2.: Observed relaxation times for NMR (this thesis) and SNMR (Schirov et al., 1991). Sediment type
Rel. Time, T2∗ [ms] SNMR
Rel. Time, T2∗ [ms] NMR
Rel. Time, T2 [ms] NMR
Rel. Time, T1 [ms] NMR
clay silt sandy clays clayey sands, very fine sands fine sands medium sands coarse and gravelly sands gravel deposits surface water bodies
