Institute of Environment & Resources

Britt Stenhøj Baun Christensen

Using X-ray Tomography and

Lattice Boltzmann Modeling to

Evaluate Pore-scale Processes

in Porous Media

Using X-ray Tomography and

Lattice Boltzmann Modeling to

Evaluate Pore-scale Processes

in Porous Media

Britt Stenhøj Baun Christensen

Ph.D. Thesis, May 2006

Institute of Environment & Resources

Technical University of Denmark

Using X-ray Tomography and Lattice Boltzmann Modeling to Evaluate Pore-scale Processes in Porous Media Cover: Torben Dolin & Julie Camilla Middleton Printed by: DTU tryk Institute of Environment & Resources ISBN 87-91855-13-6 The thesis will be available as a pdf-file for downloading from the institute homepage on: www.er.dtu.dk Institute of Environment & Resources Library Bygningstorvet, Building 115, Technical University of Denmark DK-2800 Kgs. Lyngby Phone: Direct: (+45) 45 25 16 10 (+45) 45 25 16 00 Fax: (+45) 45 93 28 50 E-mail: library@er.dtu.dk

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Preface The present thesis “Using X-ray tomography and lattice Boltzmann modeling to

evaluate pore-scale processes in porous media” has been submitted as a part of the requirements for the Ph.D. degree at the Technical University of Denmark (DTU). The study was performed at the Institute of Environment & Resources (E&R) at DTU from April 2001 to November 2005.

My advisors Assistant Professor Dorthe Wildenschild, Oregon State University,

Research Scientist Marcel Schaap, GEBJ Salinity Laboratory, and Professor Karsten Høgh Jensen, University of Copenhagen are thanked for their help and guidance throughout the study. Especially thanks to Dorthe Wildenschild for introducing me to the fascinating yet occasionally frustrating world of experimental work using X-ray tomography. I enjoyed and appreciated working in the laboratory with Dorthe Wildenschild and former Ph.D.-student Katherine A. Culligan. A special thanks also goes to Marcel Schaap without whom the modeling part of my Ph.D.-project could not have been done. Marcel Schaap always showed great patience and willingness to discuss questions and help with any modeling problems, for this I am very grateful. I would also like to thank former Ph.D.-student Petri de Willigen for a very beneficial collaboration on modeling issues, which I profited greatly from and helped keep me on track.

The first part of the thesis gives an introduction to the subjects relevant to my

Ph.D.-project and an overview of what I have worked with during my study. It summarizes the main results gained during the project and introduces the remainder of the thesis consisting of five journal papers. The first four papers deal with some of the laboratory work that I have been involved in. All experimental work in the second, third, and fourth paper were performed in very close collaboration with the first and second authors of the papers. These three papers have all been published or submitted for publication, whereas the first paper has not been submitted for publication yet. The last paper presents work using numerical modeling and is in a draft format not yet submitted for publication.

The Technical University of Denmark funded the study.

Copenhagen, May 2006

Britt Stenhøj Baun Christensen The papers are not included in this www-version but may be obtained from the Library at the Institute of Environment & Resources, Bygningstorvet, Building 115, Technical University of Denmark, DK-2800 Kgs. Lyngby (library@er.dtu.dk)

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Abstract A combination of experimental work and numerical modeling was used to

investigate pore-scale two-fluid-phase flow in sand and glass bead porous media systems. The experimental work made use of the non-intrusive and non-destructive imaging technique X-ray computed tomography (CT) to visualize and quantify the experimental pore-scale systems. Both a medical CT-scanner and a high resolution synchrotron based scanner system were employed. Numerical modeling of the pore-scale processes was done using a lattice Boltzmann model.

Using a medical CT-scanner and gravimetric measurements three different techniques for achieving full saturation as initial condition in a sand sample were evaluated. It was found that venting the sample with carbon dioxide prior to saturation significantly improved initial saturation levels whereas the use of degassed water and application of vacuum did not result in considerable improvements of the initial saturation.

The applicability of a synchrotron based X-ray microtomography system for studying pore-scale processes was explored by conducting drainage and imbibition experiments with sand and glass beads as porous media. The developed experimental setup and subsequent analysis of the obtained high resolution images made it possible to gain qualitative and quantitative information on different pore-scale features. For air-water and oil-water experiments fluid saturations and fluid-fluid interfacial areas were derived from the experimental images. The obtained experimental data can provide a useful data base for aiding theoretical development on pore-scale processes, but the data can especially assist in the development and testing of pore-scale models.

The oil-water-glass bead experiments were used as benchmark for testing a Shan-Chen lattice Boltzmann model. Using simple well-known two-fluid systems a calibration procedure was outlined for identifying the dimensionless model parameters and linking them to the interfacial tension and contact angle properties of the physical system. The scaling between lattice units and physical units was addressed. By performing displacement simulations in a well-defined system the calibrated model was seen to be able to produce realistic capillary pressures comparable to the pressures observed in the experimental system. Initial attempts to perform simulations of the oil-water experimental setup were carried out, but no reliable results were obtained due to boundary condition problems and large simulations times.

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Resumé En kombination af eksperimentelt arbejde og numerisk modellering blev

anvendt til undersøgelse af poreskala to-fase strømning i porøse medie systemer af sand og glasperler. Det eksperimentelle arbejde gjorde brug af den ikke-indtrængende og ikke-destruktive røntgenstråle billedmetode CT-teknik (Computed Tomography) til at visualisere og kvantificere de eksperimentelle poreskala systemer. Både en medicinsk CT-scanner og et synkrotron baseret skanningssystem med høj billede opløselighed blev anvendt. Numerisk modellering af poreskala processerne blev gjort ved hjælp af en lattice Boltzmann model.

Ved anvendelse af en medicinsk CT-scanner og gravimetriske målinger blev tre forskellige teknikker til at opnå fuld mætning som begyndelsesbetingelse i en sandprøve evalueret. Det blev vist, at ventilering af prøven med kuldioxid inden opfugtning signifikant forbedrede det initielle mætningsniveau, hvorimod anvendelse af afgasset vand og påføring af vakuum ikke resulterede i betydelige forbedringer af den initielle mætning.

Anvendeligheden af et synkrotron baseret røntgenstråle mikrotomografi system til at studere poreskala processer blev undersøgt ved at udføre drænings- og opfugtningsforsøg med sand og glasperler som porøse medier. Den udviklede forsøgsopstilling og efterfølgende analyse af de opnåede billeder af høj opløselighed gjorde det muligt at få kvalitativ og kvantitativ information om forskellige poreskala egenskaber. For luft-vand og olie-vand forsøg blev mætningsgrader og overfladeareal mellem faserne udledt fra de eksperimentelle billeder. De opnåede forsøgsdata kan udgøre en nyttig database til hjælp med udvikling af teorier for poreskala processer, men dataene kan især fremme udvikling og testning af poreskala modeller.

Olie-vand-glasperle forsøgene blev brugt som udgangspunkt for testning af en Shan-Chen lattice Boltzmann model. Ved anvendelse af simple veldefinerede to-fase systemer blev en kalibreringsprocedure skitseret til identificering af de dimensionsløse modelparametre og deres kobling til overfladespænding og kontaktvinkel egenskaberne af det fysiske system. Det blev taget fat på skaleringen mellem lattice enheder og fysiske enheder. Ved at udføre simuleringer af væskefortrængning i et veldefineret system sås den kalibrerede model at være i stand til at frembringe realistiske kapillartryk sammenlignelige med trykkene observeret i det eksperimentelle system. Indledende forsøg på at udføre simuleringer af olie-vand forsøgsopstillingen blev gennemført, men pga. problemer med grænsebetingelser og lange simuleringstider blev ingen pålidelige resultater opnået.

Table of contents 1. Introduction .............................................................................................................. 1 2. Objectives ................................................................................................................. 4 3. X-ray computed tomography for multiphase flow in porous media ........................ 4

3.1 X-ray computed tomography (CT) ................................................................... 4 3.2 Medical CT scanner experiments ..................................................................... 5

3.2.1 Initial saturation experiments ................................................................... 5 3.2.2 Flow rate dependence experiments .......................................................... 6

3.3 Synchrotron based CMT experiments .............................................................. 7 3.3.1 Sand experiments...................................................................................... 7 3.3.2 Air-water glass bead experiments ............................................................ 8 3.3.3 Oil-water glass bead experiments............................................................. 8

4. Numerical modeling of pore-scale processes in porous media ................................ 9 4.1 Lattice Boltzmann (LB).................................................................................... 9 4.2 LB simulations................................................................................................ 10

4.2.1 Model calibration.................................................................................... 10 4.2.2 Simulation of oil-water experiments ...................................................... 11

5. Conclusions ............................................................................................................ 13 6. Perspectives ............................................................................................................ 14 References ...................................................................................................................... 16 Appendix A - Identification of LB parameters for an air-water system......................... 25 Papers

I. Using X-ray computed tomography to evaluate the initial saturation resulting from different saturation procedures

II. Estimating multi-phase pore-scale characteristics from X-ray tomographic data

using cluster analysis-based segmentation

III. Interfacial area measurements for unsaturated flow through a porous medium

IV. Pore-scale characteristics of multiphase flow in porous media: A comparison of air-water and oil-water experiments

V. Calibrating the Shan-Chen lattice Boltzmann model for immiscible displacement

in porous media

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1. Introduction Problems involving multiple-fluid-phase (hereinafter multiphase) flow processes

in porous media play a key role in a variety of different science and engineering disciplines such as those dealing with hydrology, petrophysics, and chemistry. One of the most important topics in environmental studies concerns water resources management and here multiphase flow is of great importance in relation to both unsaturated flow and transport, and contaminated flow in the subsurface.

Usually investigations are carried out at the macro-scale, which is reasonable considering the scale of the problem being explored. However, experimental studies on characterization of soil hydraulic properties have observed phenomena that cannot be explained by the conventionally used macroscopic multiphase flow equations based on extrapolations of the empirical Darcy’s law. Traditionally, flow in unsaturated porous media is assumed to be adequately characterized by the capillary pressure-saturation relationship and the unsaturated hydraulic conductivity characteristic. Normally, these characteristics have been presumed to be independent of the flow conditions, i.e., they are identical regardless of the flow being steady state or transient and unaffected by measurement conditions being in equilibrium or non-equilibrium. An increasing amount of literature has reported on experimental observations that are not consistent with these assumptions. Already in the 1960’s and early 1970’s work was published showing the hydraulic properties to be affected by dynamic effects. Davidson et al. (1996) among other things observed that more water was retained in a sample if several pressure steps instead of just one pressure step were used to drain a sample. Contrary to these findings several other scientists observed the opposite phenomenon. In drainage experiments by Topp et al. (1967) no difference was observed between static equilibrium and steady state retention curve results, whereas transient retention curve experiments resulted in higher water content at a given capillary pressure. In addition, more water was retained when increasing the flow rate. Results by Smiles et al. (1971) corroborated the tendency of the retention curve to move upwards under transient drainage conditions. Experiments by Vachaud et al. (1972) also confirmed these results, showing similar dynamic effects of the drainage curve for both multi-pressure-step and one-pressure-step experiments, with more water being retained in the one-step experiments. However, despite all these investigations, no certain conclusions on the causes of the dynamic effects were drawn. Within the last decade researchers have resumed work on the topic. The results of the later work generally show an upward shift in the retention curve under transient drainage conditions, with the effect being more pronounced at high flow rates and in experiments where few pressure steps are used for draining the sample (Plagge et al., 1999; Hollenbeck and Jensen, 1999; Mortensen et al., 2001; Wildenschild et al., 2001). Although, the results have given rise to several suggestions on the causes of the observed effects, conclusive results are still lacking.

The problems observed at the macro-scale have prompted interest in understanding the processes going on at the smaller scale, the pore-scale. Instead of continuously extending equations that are based on empirical relationships and intended

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for specific purposes (like Darcy’s equation developed for single phase flow in porous media), the intention is to gain an understanding of the fundamental pore-scale processes and thereby be able to understand and describe the governing macro-scale processes. A long line of work performed by especially Gray and Hassanizadeh (e.g., Hassanizadeh and Gray, 1993; Gray and Hassanizadeh, 1998; Gray, 1999) has been carried out to derive constitutive equations describing flow at the pore-scale and how to scale to larger scales (see review of Gray and Miller (2005) and Miller et al. (1998) and the references herein).

Developments of new experimental techniques have allowed us to not only observe things we could only guess at before, but also to collect quantitative information. Scanning electron microscope images (SEM) of thin cross sections of soil samples have been used to visualize soil structure and for estimating characteristic soil properties such as specific surface area, porosity and permeability (Blair et al., 1996; Schaap and Lebron, 2001). The SEM technique is normally limited to two-dimensional representation of the pore space and furthermore the sample is destroyed in the process. Construction of micromodels makes it possible to visualize two-dimensional pore-scale processes in an artificial and idealized porous medium while keeping the sample intact for further studies. Micromodels have among others been used to study three-fluid capillary pressure saturation relationships (Soll et al., 1993), nonaqueous phase liquid (NAPL) movement (Keller et al., 1997; Zhong et al., 2001; Chomsurin and Werth, 2003), and colloid transport (Baumann and Werth, 2004). For studying multiphase flow in natural porous media two-dimensional light transmission methods (LTM) have been applied (Darnault et al, 2001; Mortensen et al., 2001). Two-dimensional systems will inevitably be influenced by boundary effects and three-dimensional techniques are thus often desirable for obtaining realistic information on physical behavior in porous media. Montemagno and Gray (1995) developed photoluminescent volumetric imaging (PVI) for visualizing three-dimensional multiphase flow and transport. Fredrich (1999) adopted the laser scanning confocal microscopy (LSCM) technique to image the three-dimensional microgeometry of porous materials. More readily available are the non-intrusive and non-destructive imaging techniques magnetic resonance imaging (MRI) and X-ray computed tomography (CT). Applications of MRI include studies of NAPL dissolution (Zhang et al., 2002), NAPL removal in a three-phase system (Chu et al., 2004), transport of heavy metals (Nestle et al., 2003), and colloid transport (Baumann and Werth, 2005). X-ray CT scanning has been used in a variety of applications, e.g., studies of changes in soil water distribution (Hopmans et al., 1992), air distribution patterns during air sparging (Chen et al., 1996), characterization of fractures (Montemagno and Pyrak-Nolte, 1999; Bertels et al., 2001), and observation of displacement mechanisms (Mogensen et al., 2001). When a synchrotron X-ray source is used, high resolution computed microtomography (CMT) images can be obtained. CMT allows for observation and quantification of pore-scale process and has been used to study, e.g., pore space structure and connectivity (Lindquist and Venkatarangan, 1999; Seidler et al., 2000; Al-Raoush and Willson, 2005a) and various multiphase flow processes (Auzerais et al., 1996; Coles et al., 1998; Bauters et al., 2000; Al-Raoush and Willson, 2005b).

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Significant advances in computational abilities have furthermore sparked considerable progress in development of pore-scale modeling tools. Numerical models such as the smoothed particle hydrodynamics (SPH) method (e.g., Morris et al., 1997) and the pore-morphology approach (Hilpert and Miller, 2001) have been developed for porous medium systems. However, the pore-scale models having received most attention within recent years have been pore network models and lattice Boltzmann (LB) models. Network models have been widely applied for multiphase flow problems, e.g., Mogensen and Stenby (1998) developed a dynamic two-phase network model and looked at the sensitivity of residual oil saturation to contact angle, flow rate, and pore structure in investigations of imbibition. Held and Celia (2001a) performed network model simulations of mass transfer and relationships between capillary pressure, saturation, interfacial area, and common lines (Held and Celia, 2001b). Celia et al. (1995) provides an overview of the work done in the early 90’s whereas Blunt (2001) and Blunt et al. (2002) provide more recent reviews of pore network applications. The LB method has been used to examine numerous fluid flow phenomena and many studies have applied LB modeling to complex geometries such as porous media. Studies include modeling of e.g., solute transport in variably saturated porous media (Zhang et al., 2002), NAPL dissolution (Knutson et al., 2001), and modeling of bacterial chemotaxis (Hilpert, 2005). Several permeability investigations have shown good qualitative agreement with experimental results (Koponen et al., 1998; Maier et al., 1998; Zhang et al., 2005). Furthermore, Pan et al. (2004) performed LB simulations of the capillary pressure-saturation relationship in a randomly generated sphere pack and likewise found good qualitative agreement with laboratory measurement.

Whereas e.g., network models use simplified geometries for illustrating the network of pores in a porous media, the LB model is capable of incorporating the exact pore structure. If detailed experimental data showing the complex geometry of the porous medium is available from e.g., X-ray CMT experiments, the LB model is an obvious choice for modeling multiphase pore-scale flow. Ferréol and Rothman (1995) looked at LB simulations of singe- and two-phase flow in a porous medium constructed from X-ray CMT images of Fontainebleau sandstone. Likewise, Hazlett et al. (1998) used a pore network derived from X-ray CMT images as input in a LB model; in this case for studying the effect of wettability and flow conditions on relative permeability in immiscible displacement simulations. In a soil geometry based on X-ray CT images of a sandy material Krafczyk et al. (2000) performed LB simulations of a drainage and imbibition curve that qualitatively compared well with experimental data. Mantle et al. (2001) looked at single-phase LB simulations in a porous medium obtained from MRI images and found good agreement between simulations and experiments when comparing velocity distributions. Also LSCM images of Berea sandstone (O’Connor and Fredrich, 1999) and PVI images of crushed glass beads (Zhang et al., 2000) have been used as input for LB simulations.

Using a pore structure obtained from X-ray CMT images of a sintered glass porous medium as input, Vogel et al. (2005) compared the ability of a network model, a morphology based model and a LB model to determine the capillary pressure-saturation relationship for primary drainage. Even though the three models gave consistent results,

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the far more complex, but also computationally demanding, LB approach was concluded to have most potential and hence to be a promising tool for detailed investigations of multiphase phenomena. It should be noted that Vogel et al. (2005) did not make a direct comparison with experimental data.

Most often the reported LB studies on multiphase phenomena in porous media have performed a qualitative comparison of simulations and experimental data. No studies have been reported on detailed quantitative comparison of multiphase LB flow simulations and laboratory observations using the exact same three-dimensional porous medium.

2. Objectives The overall objective of this study has been to advance our understanding of the

processes controlling multiphase flow in porous media at the pore-scale. For this purpose both experimental work using X-ray CT and pore-scale modeling using a lattice Boltzmann model have been performed and the usefulness of these relatively novel experimental and numerical modeling tools for studying multiphase flow processes has been explored. A specific goal of the study has been to conduct highly accurate drainage and imbibition experiments and hereby provide a detailed data base on porous media characteristics and fluid behavior on the pore scale. The data base provides experimental support for evaluation and further development of pore-scale theories and numerical models.

In the following the used methods are briefly presented together with the main results. A more detailed description of the bulk part of the performed work has been reported in five journal papers enclosed in this thesis.

3. X-ray computed tomography for multiphase flow in porous media

3.1 X-ray computed tomography (CT)

Although, originally developed and applied in the medical sciences the X-ray CT technique is now being used within many different disciplines for various purposes due to its ability to provide three-dimensional information of an object without destroying the object in the process. Basically, electromagnetic radiation is passed through the object of interest and the amount of radiation attenuated (absorbed) in the object is recorded and compared to the incident radiation. The attenuation depends on the density and the atomic composition of the scanned material and the photon energy of the X-ray beam.

In earth science research different kinds of scanner systems have been applied. The conventional medical scanners were first developed and have the advantage of being widespread and thoroughly tested and thus rather easily accessible for

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experimental investigations. The medical scanners are optimized for examining human tissue while minimizing the given X-ray dose. This limits the applicability of the medical scanners. Industrial CT scanners have thus been developed and tailored to fit needs for e.g., higher energy levels, higher intensities and longer exposure times; among others resulting in higher image resolutions. Especially, synchrotron radiation emitted by electrons moving in a circular orbit controlled by a magnetic field in a particle accelerator has these qualities. For medical CT systems a spatial resolution of a few hundred microns is common, where as a synchrotron based system can give resolutions of a few microns per image voxel (Wildenschild et al., 2002). CT-scanner systems are often characterized as belonging to different ‘generations’ based on how the X-ray source and detector system is arranged and how the scanning is performed.

All scanner systems essentially produce a polychromatic beam, meaning X-rays with a range of photon energies. However, if synchrotron radiation is used it is of such high intensity that a monochromator can be used to limit the radiation to a single energy level. By doing so the attenuation of X-rays by specific elements in a sample can be controlled and it is hence possible to enhance the phase contrast in the images.

A thorough description of the CT technique including the physical background, potential error sources and example applications can be found in Ketcham and Carlson (2001), Kinney and Nichols (1992), and Clausnitzer and Hopmans (2000). A discussion of different scanning systems including some of their advantages and limitations is given in Wildenschild et al. (2002); the discussion includes a specific description of the two systems used in this study.

3.2 Medical CT scanner experiments

3.2.1 Initial saturation experiments

In this study a medical CT scanner was used for analyzing different laboratory techniques commonly used for achieving full saturation of porous media samples. In most laboratory techniques used for measuring unsaturated hydraulic properties or for analyzing unsaturated flow behavior it is important to achieve a state of full water saturation as initial condition and avoid the effect of entrapped air. Three laboratory techniques were evaluated and tested for their ability to improve the saturation procedure of a porous sample: (1) venting of the sample with carbon dioxide prior to saturation, (2) applying vacuum to the sample in the beginning of the saturation procedure, and finally (3) using degassed water for saturating the sample. Before useful results could be obtained several adjustments of the experimental setup were performed. Preliminary experiments showed that accurate positioning of the sample holder was particularly important for achieving reliable and comparable results. The three saturation techniques were tested as five different combinations, each performed twice on the same packing of a fine silica sand. The same general procedure was followed for each saturation test. The sample was thus allowed to saturate for the same length of time

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(using about 4.5 pore volumes of distilled water) and subsequently drained using pressurized nitrogen.

Evaluation of the different techniques was done by recording the weight of the sample and by scanning the sample in 1 mm intervals over the height using a forth-generation medical CT scanner. Analysis of the acquired X-ray attenuation values made it possible to quantify the distribution of the water saturation over the height of the sample. The gravimetric measurements and CT scanner results both showed that venting the sample with carbon dioxide prior to saturation significantly improved the initial saturation and resulted in almost full saturation of the sample. Application of vacuum and the use of degassed water did not have a notable effect on the saturation procedure and gave poor results similar to the ones where no specific saturation technique had been applied.

A detailed description of the experiments can be found in Christensen et al. (2005a).

3.2.2 Flow rate dependence experiments

In the second study where a medical CT scanner was used a number of outflow experiments were performed to investigate the occurrence of flow rate dependence of the retention curve. The effect of different grain size distributions was tested using three types of sand of varying uniformity and coarseness. An experimental setup similar to the one used in the initial saturation experiments was utilized. Retention curves based on a point measurement of the capillary pressure inside the sample and average saturation degrees calculated from the measured outflow were compared for one-step experiments with high boundary pressures and semi-static syringe pump experiments with very low outflow rates. To visualize and monitor the changes in phase distribution within the sample CT scans where taken during some of the outflow experiments.

For all three sand types positive vertical shifts of the retention curve similar to the ones previously reported in the literature were observed during the dynamic experiments. However, numerical simulations of the experiments revealed that the positive shifts most likely were caused by a lack of continuity of the air phase. Hence, the basic assumption for obtaining the capillary pressure and thereby the retention curve was violated. Analysis of the CT scanner results helped clarify the outflow patterns within the sample and identified edge effects caused by a non-uniform packing of the sand, resulting in uneven drainage of the sand matrix. In reality the packing thus resulted in a sample consisting of two distinct porous media, the inner sand core and the sand at the edge of the sample holder.

The predominant part of the research was carried out by Majken C. Looms as MSc thesis work under the guidance of Ph.D.-student Britt S.B. Christensen. Further details can thus be found in Looms (2003).

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3.3 Synchrotron based CMT experiments

Through access to the GSECARS1 bending magnet beam-line at the Advanced Photon Source (APS), Argonne National Laboratory, several experimental runs were performed where the applicability of X-ray CMT to explore and describe pore-scale characteristics of two-fluid-phase systems was studied.

3.3.1 Sand experiments

Previous experimental runs at the APS had revealed several issues regarding the experimental setup (Wildenschild et al., 2002; Wildenschild et al., 2005). To distinguish water from air in the resulting images, the water phase had to be doped (KI was used as dopant). By doing so phase contrast due to differences in X-ray attenuation was obtained. The previous experiments had also illustrated the dilemma between obtaining a high resolution for resolving details and reaching a representative elementary volume (REV) so the identified system properties are scale independent. Scanning of small samples will produce detailed images, but result in lack of a REV. This problem was overcome using a coarse grained sand as porous medium and lower image resolution. By doing so both detailed images and a REV was obtained. The lower image resolution furthermore, results in a larger photon flux through the imaged area, giving a better signal-to-noise ratio.

The coarse sand (8-20 sand) had a d50 of 0.58 mm and was loosely packed in a 6.0 mm inner diameter cylinder. Atmospheric air could freely enter and leave the sample through the top of the cylinder, where as water was connected to the sample through a porous semi-permeable membrane at the bottom of the cylinder, allowing water but not air to pass the membrane. A finely graduated pipette functioned as water reservoir and at the same time made it possible to estimate the change in saturation in the sample. By stepwise lowering or raising the pipette the capillary pressure was altered and a primary drainage and secondary imbibition of the sample was performed. Scans were taken for each change in saturation. With an image resolution of 17 microns per voxel it was possible to observe pore-scale elements such as local differences in saturation patterns during drainage and imbibition of the sand. Looking at the air phase configuration showed that the air phase was distributed as larger bubbles on imbibition compared to drainage.

The experiments have been reported in Wildenschild et al. (2005) where also a thorough explanation of the techniques used for the image analysis is presented. The analysis of the images was very troublesome and different approaches were examined. Although the image analysis may still need improvements, the procedure outlined in Wildenschild et al. (2005) has proven successful and has thus been applied to all APS results reported herein.

1 GeoSoilEnviro Consortium for Advanced Radiation Sources

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3.3.2 Air-water glass bead experiments

Using sand as the porous medium resulted in several problems with the analysis of the images. The presence of different minerals in the sand grains resulted in faulty high X-ray absorption values, making it difficult to distinguish the sand from water. In addition, the highly irregular shape of the sand grains made it difficult to accurately outline the geometry of the sand. To eliminate these problems the next experiments were performed using glass beads.

A series of drainage and imbibition cycles were performed on an air-water soda lime glass bead system. The diameter of the glass beads ranged between 0.6-1.4 mm. The beads were loosely packed in a cylinder with an inner diameter of 7.0 mm. Air could enter and leave the sample trough the top of the cylinder that was open to the atmosphere. The water reservoir was connected to a syringe pump used to induce different flow rates and directions in the system. A pressure transducer connected to the water reservoir measured the water phase pressure. The sample holder was mounted on a stage that was rotated in the X-ray beam in small increments resulting in 360 frames for each scan recorded on a CCD camera. A resolution of 17 microns per voxel was achieved for the imaged 5 mm vertical section of the sample.

Subsequent image analysis made it possible to quantify various properties of the system, including fluid saturations and fluid-fluid interfacial areas. Results of the air-water interfacial area versus water saturation showed an increase in interfacial area as saturation decreased until a maximum was reached at saturations of 20-35 %, after which the interfacial area decreased as the saturation continued to zero. The results agreed well with numerical studies as well as theoretical predictions reported in the literature.

A complete description of the experimental setup and results can be found in Culligan et al. (2004).

3.3.3 Oil-water glass bead experiments

A setup similar to the one used in the air-water experiments was applied to study an oil-water system. The main differences in the setup being attributed to the requirements of handling of the oil, and that the glass beads were sintered together before being placed in the sample cylinder. This was done to prevent the beads from shifting during the experiments. Shifting of the porous medium had previously resulted in problems with obtaining a representative image of the position of the porous medium. Soltrol 220, a light nonaqueous phase liquid (LNAPL) dyed red with small amounts of Oil Red O, was used as a non-wetting oil phase. A constant pressure oil reservoir was connected to the top of the sample. Like for the air-water system the experiments resulted in three-dimensional images with a resolution of 17 microns per voxel.

The results of oil-water interfacial area and water saturation measurements were compared to the air-water data. The oil-water interfacial area versus saturation curves for different drainage and imbibition cycles were more uniform than was the case for

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the air-water system. The oil-water interfacial area values were generally higher than the air-water values. This is likely related to the difference in interfacial tension between the two systems.

A complete description of the experimental setup and results can be found in Culligan et al. (2005).

4. Numerical modeling of pore-scale processes in porous media

4.1 Lattice Boltzmann (LB)

In recent years interest in the lattice Boltzmann (LB) method has been rapidly growing. The LB method was originally developed as an improvement to the lattice gas (LG) method (Frisch et al., 1986). Today, the LB method is being applied in many different areas dealing with computational fluid dynamics (CFD). Recent reviews of the LB method and its application can be found in e.g., Chen and Doolen (1998), Házi et al. (2002), Nourgaliev et al. (2003), and Raabe (2004). The method is appealing because of its conceptual simplicity and many believe it has the potential to overcome some of the problems encountered with the conventional continuum-based CFD methods (such as the incorporation of geometrically complex boundary conditions) and as such become a better modeling tool. New variations on the method are still being formulated and improvements to existing versions are continuously developed.

The most commonly used types of LB models for multiphase flow modeling are the ones normally referred to as the ‘color’ model, the ‘pseudo-potential’ model, and the ‘free-energy’ approach. The first multi-component LB model to appear was a so-called color model developed by Gunstensen et al. (1991) which was based on work by Rothman and Keller (1988), who where the first to extend LG to model two immiscible fluid components. The model was later extended by others such as Grunau et al. (1993). The model labels the component particles using different colors and the applied collision rules enable modeling of interfacial tension. One of the serious limitations of the model is the lack of thermodynamic background (Házi et al., 2002). The so-called pseudo-potential LB method is probably the most widely used of the three methods. It is also referred to as the Shan-Chen model as it was originally developed by Shan and Chen (1993). Several improvements and extensions to the original model have later been made (e.g., Shan and Chen, 1994; Shan and Doolen, 1995; Martys and Chen, 1996; Martys and Douglas, 2001). The model introduces microscopic interaction potentials between fluid particles to induce forces in the system. Due to the interaction forces the momentum is not conserved locally at each lattice node, though it will be conserved for the system as a whole (Shan and Chen, 1993). This results in the unfortunate presence of spurious currents at interfaces (Nourgaliev et al., 2003). Compared to the color model, the problem of spurious currents, and thereby instability issues, is far less in the Shan-Chen model (Hou et al., 1997). Like the color model, the Shan-Chen model is not based on thermodynamically sound assumptions, making for

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instance a correct definition of temperature difficult (Házi et al., 2002; Nourgaliev et al., 2003). This makes the two aforementioned models mostly suited for isothermal multi-component flow investigations (Chen and Doolen, 1998). In contrast, the free-energy approach developed by Swift et al. (1995, 1996) incorporates better interfacial thermodynamics, for example allowing for a well-defined representation of temperature. Regardless of further improvements of this method by others, the free-energy approach also has several drawbacks (Házi et al., 2002; Nourgaliev et al., 2003). Other LB methods, such as the so-called thermal model, are under development and some show promise to overcome several of the problems of earlier versions by building on a more realistic and physics-based framework and improving the stability properties.

The work presented in this thesis was carried out using a LB model based on the Shan-Chen method. A detailed description of the model is given in Christensen et al. (2005b).

4.2 LB simulations

The detailed X-ray CMT data obtained in this study provide a suitable data base for testing the validity of numerical pore-scale models such as the LB model. Using the oil-water-glass bead experiments as benchmark, a newly developed LB code was calibrated and tested.

4.2.1 Model calibration

Before LB simulations of a system similar to the experimental setup can be carried out, suitable LB model parameters have to be determined. Issues such as how to identify appropriate LB parameters and how to most accurately scale the simulations to comply with physical data are rarely addressed in the literature. A significant portion of the literature on LB modeling deals with theoretical investigations concerning fundamental physics. In addition, qualitative results have often been the primary focus making it unnecessary to relate model parameters and units to physical quantities. In Christensen et al. (2005b) a procedure for relating model parameters to physical characteristics of the experimental system is proposed.

As the main interest here is investigating flow of two immiscible fluids in a glass bead porous media system, the main model parameters that need to be identified are the fluid-fluid and fluid-solid interaction parameters. The fluid-fluid interaction parameter, Gc, controls the cohesion between fluids and is thus linked to the interfacial tension. The fluid-solid interaction parameter, Ga, determines the wetting (adhesion) properties of the fluids and consequently defines the contact angle at the intersection of fluids and solids. The model parameters are determined via simulations of simple, well-defined two-fluid systems and by applying Laplace’s law (see e.g., Bear, 1972 or Dullien, 1992) as the basic equation.

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Simulations of a range of spherical droplets with different radii resulted in feasible values for the fluid-fluid interaction parameter and thereby also determination of the interfacial tension in lattice units. In accordance with the wetting properties of the experimental system, a fluid-solid interaction parameter was determined so a zero degree contact angle also was achieved in the LB simulations. Two different types of simulations were used to estimate the fluid-solid interaction parameter: Simulations of a droplet of wetting fluid placed on a solid surface surrounded by non-wetting fluid gave an indication of an appropriate parameter value. However, due to periodic boundary conditions influencing the results, simulations using a capillary tube partly filled with wetting and non-wetting fluid were subsequently performed and a value for the fluid-solid interaction parameter was estimated more accurately.

In the LB model all variables are defined in dimensionless lattice units. For comparison with the experimental data in physical units, scaling factors relating lattice and physical units are needed for transforming the LB data into meaningful quantities. For the static oil-water data it was only necessary to scale space. This was done using the voxel resolution of the images as the scaling parameter. Having defined the scaling of space and knowing the physical and lattice interfacial tensions a relation between physical capillary pressure and lattice capillary pressure was established.

The calibrated model was tested by performing a series of displacement simulations in a setup with a slit. The lattice capillary pressure from the displacement simulations was converted into physical units and the agreement with expected capillary pressure for different pore sizes was found to be reasonable. Furthermore, the results showed that the calibrated model allowed for simulations of a pressure range similar to the one observed in the oil-water experiments.

For the oil-water system an analysis of the forces acting on the flow system showed that the capillary forces dominated over the gravitational and viscous forces justifying that these latter two forces could be neglected in the simulations. Contrary for an air-water system, the effect of gravity would be significant and should therefore be considered in the LB model by including the gravitational parameter. This parameter would thus also need to be identified. A short description of how this might be done is given in appendix A.

4.2.2 Simulation of oil-water experiments

The obvious next step after having calibrated the LB model to the physical properties of the oil-water system was to perform simulations of the experimental setup using the identified LB parameters. Several simulation attempts were conducted, but various problems with the setup of the model and very long simulation times meant that only preliminary results that did not allow for a reasonable comparison to the experimental data was obtained. In the following, a short description of the LB simulation attempts is presented.

The size and shape of the simulated domain was defined to represent the experimental conditions as accurately as possible, and to capture the behaviour observed

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in the acquired experimental data. The original three-dimensional imaged data set had the dimensions of 650x650x358 voxels. In the horizontal xy-plane this imaged volume included the porous sample as well as the sample holder and some surrounding air. The subsequent analysis of the experimental data (Culligan et al., 2005) was done on a smaller volume (266x301x300) cropped from the original scanned volume and consisting only of porous media. (The cropped volume comprises the largest cube that could be fitted inside the cylindrical sample). In this manner boundary effects were minimized, in particular problems with the sintered glass bead column not fitting tightly inside the sample holder could be diminished.

However, for the LB simulations the specification of the boundary conditions required particular consideration. By using a domain similar to the cubic domain used for analysis of the experimental data the LB boundary conditions would be difficult to define and the results would thus be affected by the boundaries. In the horizontal directions it was therefore chosen to include the full original circular sample holder as a solid wall, which thus functioned as a no-flow boundary condition similar to the experimental system. For locating the exact position of the glass beads and the sample holder the experimentally obtained images of the dry sample were used. Initialization of the LB model was done using image files defining the LB domain and the position of solids. The number of pixels or voxels in the image files defined the number of lattice units in the domain. For the oil-water system the images from the experiments were used directly as input to the LB simulations, hence resulting in the same degree of detail in simulations and experiments. One lattice unit thus corresponded to a spatial resolution of 17 microns per voxel.

In the vertical direction extra layers were added in both top and bottom of the domain representing reservoirs of non-wetting and wetting fluids, respectively. These fluid reservoirs functioned as smoothing buffer layers between the boundary conditions applied in the vertical direction and the porous domain. The upper and lower boundary conditions were defined such that fluid entering and leaving the porous domain was in accordance with the conditions applied in the quasi-static imbibition and drainage experiments. In the experiments a flow boundary condition was used, but since the LB code was not yet capable of handling flow boundary conditions, the drainage and imbibition processes were driven by matching the static capillary pressure measured at the end of each equilibration period in the experiments. Thus, a constant pressure boundary condition was applied (which in the LB model is equivalent to applying constant densities). The physical capillary pressure from the experiments was converted to lattice units using the relation for scaling between lattice and physical units given in Christensen et al. (2005b). The boundary fluid densities resulting in the desired equilibrium capillary pressures were then found using results from the droplet simulations performed when calibrating the LB model for the oil-water system (Christensen et al., 2005b).

To further reduce the simulation times only a few points on an imbibtion and a drainage curve were selected and the simulations initialized using the observed fluid distribution of the first quasi-static point on the curves. Nevertheless, the number of iterations needed for reaching equilibrium between each change in pressure remained

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large and thus very long simulation times were needed. It was likewise attempted to reduce simulation times by removing layers in the vertical direction and thereby reducing the domain size. When doing so the intention of sustaining a domain large enough to facilitate a representative elementary volume (REV) had to be re-considered.

Several simulations of the oil-water system were attempted, but no reliable final results were obtained. Some of the preliminary results look encouraging, but equilibrium points on drainage and imbibition curves have not yet been reached due to the prolonged simulation times. However, problems with correctly applying the vertical boundary conditions still seem to persist. An alternative to fixing the pressure at the boundaries and then try to obtain correct fluid distributions (i.e., saturations), would be to utilize flux boundary conditions and thereby withdraw/add fluid as it was actually done in the experiments. This would fix the saturation in the domain and the capillary pressure could then be calculated when equilibrium was reached. However, equilibration times would remain equally long.

Computational limitations inevitably have to be taken into consideration when attempting simulations of a multiphase porous medium system like the current oil-water system. The LB model has the ideal architecture for parallel computing with very good scaling characteristics. The present LB code was thus also executed on multiple CPUs. Apart from a potential limitation in the number of accessible CPUs, extensive memory requirements can constitute a problem. For simulations of the oil-water system several GBs of RAM were needed. The initial simulations of the oil-water system suggested that at least 10,000 CPU hours (using CPUs running at 1350 MHz) were needed if a satisfactory number of points along a retention curve for a domain equal to the experimentally imaged volume were to be simulated.

5. Conclusions The main conclusions of the study are: Evaluation of different techniques for initially achieving full saturation in a sand

sample showed that venting the sample with carbon dioxide prior to saturation clearly improved initial saturation whereas the use of degassed water and vacuum did not improve the saturation significantly. These findings were documented by both gravimetric measurements and CT scanning results.

Investigations of dynamic effects on the retention curve using a traditional flow cell

and outflow experiments produced results showing positive vertical shifts of the retention curve for three types of sand. After use of numerical simulations it was concluded that the observed shifts most likely could be attributed to lack of air phase continuity. Conclusions on probable causes for previously reported dynamic effects could thus not be drawn from these experiments.

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Experiments conducted at the APS using a coarse sand and glass beads as porous media led to further improvements of the experimental setup and the image analysis procedures. The drainage and imbibition experiments produced detailed qualitative insight into fluid flow patterns, and more importantly quantitative data on fluid saturations and fluid-fluid interfacial areas for both an air-water and an oil-water glass bead system were obtained. The analyzed images may serve as a data base for developing and testing pore-scale models.

The obtained results showed that the X-ray CT scanner technique gives reliable

results and has the possibility of providing additional information about the interior of the sample compared to other more traditionally used methods. Especially the use of synchrotron CMT produces images of very high resolution making it possible to obtain both qualitative and quantitative data of various pore-scale features.

A Shan-Chen LB model was tested on various physical scenarios. Parametrisation of

the LB model for simulating a physical two-phase porous medium system was outlined. The dimensionless lattice parameters were identified through simulation of simple well-known two-fluid-phase systems whereby the interfacial tension and contact angle properties of the physical system and the LB model were linked. The issue of how to go from lattice units to physical units was addressed. By performing displacement simulations in a well-defined system the calibrated model was tested. It was shown that the calibrated model was able to generate realistic capillary pressures that are within the pressure range of the experimental system. Several attempts to use the calibrated LB model to perform simulations of the specific physical system did not produce reliable results due to problems with the boundary conditions and large simulation times for reaching equilibrium.

6. Perspectives

The combination of using experimental techniques such as X-ray computed tomography (CT) and numerical tools like the Lattice-Boltzmann (LB) method shows promise for theoretical developments on pore-scale processes. Both CT and LB simulations can illustrate specific features of a physical system.

Synchrotron X-ray CMT has the ability to visualize and quantify real world characteristics such as fluid-fluid and fluid-solid configurations under various experimental conditions. Like any other experimental technique it is limited by the accuracy of the setup and post-processing of the collected data. These matters, however, have been improved significantly within recent years. The CMT method still has the drawback of being very time consuming, which in combination with limited access to user scan time restricts the applicability of the technique.

During the course of this study, parameters such as beam line stability and uniformity of the incident beam intensity improved at the APS facility, resulting in shorter scan times and better statistics on the images. Also, new camera lenses and

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improvements in the reconstruction software significantly added to the quality of the images. Synchrotron CMT has undergone rapid development in recent years, a trend that is expected to continue in the coming years leading to further improvements of the technique.

In addition to the applications shown in this study, future use of synchrotron CMT could focus on the study of e.g., the nature of fluid film flow, biological clogging of a porous medium, mass transfer across fluid-fluid interfaces or NAPL movement. All these subject areas are of significant scientific interest.

An important motivation for performing the oil-water experiments at the APS was to obtain a data set that could be used to test the LB method. Even though LB simulations require substantial simulation times, numerical simulations offer an attractive supplement to laboratory experiments for investigations of pore-scale fluid systems. As the numerical model is simulating a simplified version of the physical world, other problems than those encountered in the experimental system are present. In the LB model all variables are in lattice units and thus dimensionless, a scaling to physical units therefore often needs to be considered. This is not always trivial and further work on the topic would be useful. The work done on LB modeling highlighted some of the pitfalls and problems with applying the model to pore-scale fluid flow systems where a porous medium is present. Several issues still remain to be resolved. The calibration of the LB model did not result in a unique set of model parameters. The sensitivity of the model results to different parameter sets needs to be investigated further.

Another issue of interest in future work is that the LB simulation domain is discretized using a number of pixels (or voxels), meaning that all porous medium surfaces (except straight lines) will be represented with a staircase-like surface. Initial simulations for a domain where the resolution used for discretizing the porous medium was altered several times showed clear differences in fluid distributions for changing spatial resolution. How great an effect the chosen pixellation of the LB domain can have on the simulation results should thus be considered, especially when comparing simulations with physical data where the porous medium will have a continuous surface.

Simulations of the experimental setup have not yet turned out successfully. The application of physically correct (i.e., flux) boundary conditions needs to be examined further. The important comparison of detailed experimental data with LB simulation results thus still remains to be addressed. The experimental data is now present for such a comparison and an evaluation of the LB method as a technique for studying pore-scale processes should hence be within reach. Other pore-scale models such as network models may also be tested on the data set. A comparison with the experimental data should help reveal the significance of some of the problems identified during the calibration process and hopefully contribute with improvements to the parameterization steps.

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Appendix A - Identification of LB parameters for an air-water system

Identification of the fluid-fluid interaction parameter and hence the lattice interfacial tension for an air-water system can still be done using simulations of spherical droplets with different radii. To obtain the gravitational parameter scaling of three of the fundamental units is indirectly applied by using the three scaling parameters: ∆x for space, ∆t for time and M for mass. The gravitational parameter, Z, can be determined from the following equation using scaling parameters for the physical density difference, ∆ρ, and gravitational constant, g:

22 )()( txMnZg∆∆

∆=∆ρ (1)

∆x is defined using the resolution of the experimental images and M/(∆t)2 is found from scaling of the interfacial tension:

L

P

tM

σσ

=∆ 2)(

(2)

σP is in physical units and σL is in lattice units. The latter is found from the spherical droplet simulations. The density difference in lattice units, ∆n, can also be obtained from the droplet simulations. For a one-component LB model this has been tested and works well for identifying Z, but for a two-component LB model (which is desirable for the actual experimental systems) identification of ∆n is problematic as it changes with the radii of the simulated spherical droplets. (It should also be noted that the simulated fluids are slightly compressible and ∆n therefore depends on gravity.) An alternative approach should thus be used for obtaining ∆n.

Assuming that the gravitational parameter has been identified using equation 1, the fluid-solid interaction parameter remains to be determined. This could be done in the same way as outlined in Christensen et al. (2005b); the gravitational parameter just needs to be included in the simulations. However, to overcome some of the uncertainties especially due to the diffuse nature of the fluid-fluid interface and the simulations being singly, an alternative could be to estimate the parameter from simulations of capillary rise in tubes with different radii (examples are shown in figure 1).

26

Figure 1. Examples of capillary rise in tubes with different radii. Simulations were run until equilibrium was reached using a domain size of 100x100 lattice units.

The rise of fluid in a tube is controlled by gravitational and capillary forces:

θσπρπ cos22 RghR =∆ ⇒ R

gh θσρ cos2=∆ (3)

where θ is the contact angle, R is the radius of the capillary and h is the height difference between the fluid level inside and outside the capillary tube. Rearranging equation 3 results in:

ρθσ

∆=

gRh cos2 (4)

Introducing the physical values for the variables on the right hand side of equation 4 results in a constant value for the relationship between R and h. This can also be expressed in lattice units using the spatial scaling parameter:

2)( xhRRh LL ∆= (5)

Simulations should thus be performed where the fluid-solid interaction parameter is changed until a parameter value that produces the desirable relationship between R and h is found. In figure 2 examples of different relationships between R and h are illustrated as plots of data points falling on straight lines for different values of the fluid-solid interaction parameter.

27

0

10

20

30

40

50

60

0.00 0.02 0.04 0.06 0.08 0.101/RL

hL

Ga=+-0.0005

Ga=+-0.00045

Ga=+-0.0004

Ga=+-0.00035

Figure 2. The difference in wetting fluid level between the inside and outside of the capillary tube plotted against the inverse of the tube radius (two-dimentional simulations using Gc = 0.02 and Z = 0.001).

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