Imaging moisture flows in cement-based materials using electrical capacitance tomography

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uef.fi

PUBLICATIONS OF

THE UNIVERSITY OF EASTERN FINLAND Dissertations in Forestry and Natural Sciences

ISBN 978-952-61-3308-9

Dissertations in Forestry and Natural Sciences

DISSERTATIONS | ANTTI VOSS | IMAGING MOISTURE FLOWS IN CEMENT-BASED MATERIALS USING… | No 370

ANTTI VOSS

IMAGING MOISTURE FLOWS IN CEMENT-BASED MATERIALS USING ELECTRICAL CAPACITANCE TOMOGRAPHY

PUBLICATIONS OF

THE UNIVERSITY OF EASTERN FINLAND

The presence of moisture is required for most physical and chemical degradation processes to occur in cement- based materials. As a result, material’s ability to impede

the ingress of water is the key property describing its durability. Therefore, information on the internal moisture distribution and mass transport properties would be highly useful for assessing durability and predicting service life of cement-based structures. This thesis studies the applicability of electrical capacitance tomography (ECT) for monitoring moisture distributions

during flow of water in cement-based materials. The results indicate that ECT could be used as complementary

imaging tool for the currently used techniques to monitor moisture distributions and to help to characterizing

transport properties of cement-based materials.

ANTTI VOSS

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PUBLICATIONS OF THE UNIVERSITY OF EASTERN FINLAND DISSERTATIONS IN FORESTRY AND NATURAL SCIENCES

N:o 370

Antti Voss

IMAGING MOISTURE FLOWS IN CEMENT-BASED MATERIALS USING

ELECTRICAL CAPACITANCE TOMOGRAPHY

ACADEMIC DISSERTATION

To be presented by the permission of the Faculty of Science and Forestry for public examination in the Auditorium SN200 in Snellmania Building at the University of Eastern Finland, Kuopio, on February 7th, 2020, at 12 o’clock.

University of Eastern Finland Department of Applied Physics

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Grano Oy Jyväskylä, 2020

Editors: Pertti Pasanen, Jukka Tuomela, Matti Tedre, and Raine Kortet

Distribution:

University of Eastern Finland Library / Sales of publications julkaisumyynti@uef.fi

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ISBN: 978-952-61-3308-9 (print) ISSNL: 1798-5668

ISSN: 1798-5668 ISBN: 978-952-61-3309-6 (pdf)

ISSNL: 1798-5668 ISSN: 1798-5676

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Author’s address: University of Eastern Finland Department of Applied Physics P.O.Box 1627

70211 KUOPIO FINLAND

email: antti.voss@uef.fi

Supervisors: Associate Professor Aku Seppänen University of Eastern Finland Department of Applied Physics P.O.Box 1627

email: aku.seppanen@uef.fi Professor Marko Vauhkonen University of Eastern Finland Department of Applied Physics P.O.Box 1627

email: marko.vauhkonen@uef.fi

Associate Professor Mohammad Pour-Ghaz North Carolina State University

Department of Civil, Construction and Environmental Engineering

RALEIGH, NC 27695-7908 USA

email: mpourgh@ncsu.edu

Reviewers: Associate Professor Cristiana Sebu University of Malta

Department of Mathematics MSIDA MSD 2080

MALTA

email: cristiana.sebu@um.edu.mt Lecturer Jiabin Jin

The University of Edinburgh School of Engineering EH9 3JL EDINBURGH GREAT BRITAIN

email: jiabin.jia@ed.ac.uk

Opponent: Professor Burkan Isgor

Oregon State University

School of Civil and Construction Engineering CORVALLIS, OR 97331

USA

email: burkan.isgor@oregonstate.edu

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Antti Voss

Imaging moisture flows in cement-based materials using electrical capacitance tomography

Kuopio: University of Eastern Finland, 2020 Publications of the University of Eastern Finland Dissertations in Forestry and Natural Sciences; 370

ABSTRACT

The majority of the deterioration mechanisms affecting cement-based structures require the presence of moisture. Therefore, ingress of water into concrete is a major factor affecting the rate of deterioration mechanisms; water further enables the transport of aggressive agents that contribute to many physical and chemical degradation processes. As a result, the durability of cement-based materials is mostly dictated by the materials’ ability to impede transport of water. Therefore, information on the internal moisture distribution and mass transport properties of the materials would be of high importance in durability assessments and service life predictions. Tech- niques to visualize moisture movement and distributions in cement-based materials have a significant role in understanding mechanisms involved in water transport and more accurate estimation of the rate of water transport.

In this thesis, the focus is on unsaturated water movement monitoring using tomographic imaging modalities which are the most accurate techniques for that purpose. Previously used monitoring techniques include methods that are based on attenuation or scattering of ionizing electromagnetic radiation and flux of particles (e.g. neutron, X-ray, gamma-ray imaging) and utilizing magnetic properties (nuclear magnetic resonance (NMR) or magnetic resonance imaging (MRI)). These modalities provide high resolution information about the moisture distribution, but also have high costs for operational use and/or are limited to small-sized specimens. Electrical imaging can offer a lower cost monitoring option than the standard high resolution methods. The resolution in electrical imaging is lower, but the specimen size is not restricted. One electrical tomographic modality, electrical resistance tomography (ERT), has been applied for monitoring moisture in cement-based materials, but it can not be properly applied for surface-dry (electrically resistive) targets (i.e., materials with a low degree of saturation), because the method is based on electric current measurements. Especially, in the case of unsaturated specimens which often feature high electrical resistivity at the surface, adoption of ERT imaging is difficult due to the poor electric contact between electrodes and highly resistive surface of the target.

This thesis aims to overcome the contact problem of electrical imaging for surface- dry (electrically resistive) specimens by using another electrical imaging technique, electrical capacitance tomography (ECT). ECT enables measurements of very low moisture content materials since it uses capacitive measurements that do not necessary require any contact with the target. In ECT, electrical capacitances are measured between electrodes placed around the target, and the boundary data is used to reconstruct the internal permittivity distribution of the target. The reconstructed permittivity distribution, in turn, is directly related to moisture distribution since permittivities of cement-based materials vary as functions of the water content. The challenge in ECT, is that the estimation of spatially-varying permittivity from ca-

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pacitive boundary measurements is an ill-posed, non-linear inverse problem. This implies that the image reconstruction problem is non-unique in classical sense and highly sensitive to modeling or measurement errors. As a consequence, extra care must be directed to the mathematical modeling of the measurement process as well as to the reconstruction methods.

Previously, the use of capacitive measurements for tomographic imaging of cement -based materials has not been reported. In this thesis, the aim is to study the applicability of ECT for imaging unsaturated moisture flows within cement-based materials. Furthermore, the use of ECT reconstructions for characterizing materials with different moisture transport properties is investigated. To meet those aims, experimental setups for ECT imaging of moisture flows within cement-based materials are engineered and computational methods for the image reconstructions are developed. The results show that the ECT permittivity images provide information on the moisture movement and illustrate the spatially distributed moisture within cement-based materials. ECT images of the moisture flow can be utilized to quali- tatively distinguish between different materials, and moreover, they show potential to be used for quantitative characterization of material’s moisture transport properties. The studies indicate that ECT can be used as a complementary imaging tool that utilizes contact-free measurements to observe water absorption and moisture conditions in cement-based materials. ECT provides information that could be further used in predicting deterioration and durability of widely used cement-based materials.

Universal Decimal Classification:537.226.2, 620.193.23, 621.317.33, 624.012.4, 691.32, 691.53

INSPEC Thesaurus: materials properties; electric properties; cements (building materials);

concrete; mortar; moisture; water; moisture measurement; imaging; monitoring; tomography; capacitance measurement; permittivity; image reconstruction; inverse problems; mod- elling; numerical analysis

Yleinen suomalainen ontologia: materiaalitutkimus; rakennusaineet; sementti; betoni;

laasti; kosteus; vesipitoisuus; mittaus; mittausmenetelmät; monitorointi; kuvantaminen; to- mografia; sähköiset ominaisuudet; inversio-ongelmat; mallintaminen; numeeriset menetelmät

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ACKNOWLEDGEMENTS

This thesis work was carried out in the Department of Applied Physics at the Uni- versity of Eastern Finland during the years 2014-2019.

First and foremost, I would like to thank my supervisors for their excellent guidance and all the help throughout this thesis work. I am deeply grateful to my principal supervisor Associate Professor Aku Seppänen for all the support and constant guidance he has given me during this work. I also wish to express my gratitude to my second supervisor Professor Marko Vauhkonen for his valuable advices and ideas, and all the help I have received from him, both before and during this thesis project. I also want to thank my third supervisor Associate Professor Moham- mad Pour-Ghaz, without his assistance and expertise in cementitious materials, this project would have been so much harder to come to fruition.

Secondly, I want to thank the official reviewers of this thesis, Associate Professor Cristiana Sebu and Lecturer Jiabin Jin, for their time and effort spent going through my work, and providing me with professional and constructive feedback.

I am honored and looking forward to have Professor Burkan Isgor as my dissertation opponent.

I would like to thank all the current and former members of the Inverse Problems research group at Kuopio and all the people I have had the chance to interact with during the thesis work. I wish to thank everyone who helped me with the experimental work conducted in this thesis. Very special thanks go to Tuomo Savolainen for his invaluable work with experimental setups, measurement devices and ECT sensors. I also thank Kimmo Karhunen for providing me with the preliminary code package for the FE solver of ECT’s forward model at the early stages of this project.

Many thanks to my colleges Mikko Räsänen, Jyrki Jauhiainen, and Petri Kuusela for all the help and useful ideas. I would also like to thank my former and current col- leagues Gerardo González, Rahul Yadav, and Matti Niskanen, for your friendship and company in and outside the university. A big thank you goes also to all my friends outside the academia.

This thesis was financially supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 764810, Academy of Finland (project 270174) and Doctoral progamme in Science, Technology and Computing (SCITECO) of University of Eastern Finland. The financial support is gratefully acknowledged.

I wish to express my warmest gratitude to my parents, Mika and Maarit, and to my sisters, Heidi and Marika, for all the support and encouragement you have given without any reservations throughout my life. Finally, I owe my deepest gratitude to my beloved partner Johanna for her love, help and patience at all times. I am very fortunate to have you by my side.

Kuopio, December 19, 2019 Antti Voss

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LIST OF PUBLICATIONS

This thesis consists of an overview and the following four original articles, which are referred to in the text by their Roman numeralsI-IV.

I A. Voss, M. Pour-Ghaz, M. Vauhkonen and A. Seppänen, “Electrical capacitance tomography to monitor unsaturated moisture ingress in cement-based materials“,Cement and Concrete Research89, 158–167 (2016).

II A. Voss, N. Hänninen, M. Pour-Ghaz, M. Vauhkonen and A. Seppänen„ “Imag- ing of two-dimensional unsaturated moisture flows in uncracked and cracked cement-based materials using electrical capacitance tomography“, Materials and Structures51:68 (2018).

III A. Voss, P. Hosseini, M. Pour-Ghaz, M. Vauhkonen and A. Seppänen, “Three- dimensional electrical capacitance tomography – a tool for characterizing moisture transport properties of cement-based materials“, Materials and Design, 107967 (2019).

IV A. Voss, M. Pour-Ghaz, M. Vauhkonen and A. Seppänen, “Retrieval of the saturated hydraulic conductivity of cement-based material using electrical capacitance tomography“,Cement and Concrete Composites, In Review.

The original articles have been reproduced with permission of the copyright holders.

AUTHOR’S CONTRIBUTION

The publications selected in this dissertation are results of joint work with the supervisors and co-authors. The author was the principal writer in all the publications and conducted all the water ingress experiments and ECT measurements in these publications. The author implemented all the numerical computations using Mat- lab and HYDRUS 3D, and computed all the results in PublicationsI-IV. The finite element package for solving the ECT forward problem is an adaptation of the codes previously developed in the inverse problems group in the Department of Applied Physics.

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εh Finite element approximation of permittivity ε_i Finite element coefficient of permittivity ϕ_i Finite element basis function for permittivity q˜k Finite element coefficient for electric charges n_k Finite element basis function for electric charges u_j Finite element coefficient for electric potential φ_j Finite element basis for electric potential xiv

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A Finite element system matrix y Finite element solution vector

f Finite element data vector

M Measurement matrix

C Full set of measured ECT capacitance data Q Full set of measured ECT electric charge data ν Normally distributed measurement noise

H ECT forward model

t Time

ε_lin Linearization point

J Jacobian matrix

∆Q Difference electric charge data

∆ε Permittivity change

p₍_·₎(·) Regularization functional W₍_·₎ Weight matrix,W₍_·₎=L^T₍_·₎L₍_·₎ β_i Step size ofith iteration

d⁽ⁱ⁾₍_·₎ Search direction vector ofith iteration

B(∆ε) Quadratic barrier function for non-negativity constraint D(ε) Quadratic barrier function for permittivity constraint a Variablity in the unknown quantity

l_k Spatial correlation length

I Electric current

i Imaginary unit

ω Angular frequency

f Frequency

Vw Absorbed water volume

E Spatially integrated permittivity change θ Volumetric moisture content

θ_i Initial moisture content θs Saturation moisture content

K Hydraulic conductivity

Ks Saturated hydraulic conductivity Kr Relative hydraulic conductivity

Ψ Capillary head

ε_BG Homogeneous background permittivity ε^1D 1D discretization of permittivity

dε Derivative of permittivity with respect to length

∆z Distance between consecutive nodes in 1D discretization ε^3D 3D discretization of permittivity

M Mapping matrix

P Interpolation matrix

L Water front penetration

Ψe Effective wetting front capillary head fe Effective porosity

S Slope of a LS fitted curve

K^true_s True saturated hydraulic conductivity value in numerical studies K^ref_s Reference saturated hydraulic conductivity value

K^est_s Estimated saturated hydraulic conductivity value

∆K_s^est Error in saturated hydraulic conductivity estimate

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1 Introduction

Concrete is the most widely used material. The use of concrete as a primary building material in architectural structures, roads, bridges, dams, etc. is a common sight almost everywhere in the world [1, 2]. Concrete is a porous composite material that consist of a binding medium within which particles of aggregate of different sizes (crushed stone, gravel, sand, etc.) are embedded. The binder (cement paste) is a mixture of water and cement, and the binding property develops as a result of chemical reactions between cement and water, in a process called hydration. Cement is a dry and finely pulverized material itself and it needs water to initiate the hydration process and gain mechanical strength and stiffness. In addition to cement, water and aggregates, a fourth component, namely chemical admixtures are frequently used in modern concrete mixtures to improve some of the properties of concrete, either in the fresh or hardened state. For example, air-entraining admixtures can be used to enhance the durability of concrete in cold weather or the workability of fresh concrete mixture can be improved by adding water-reducing admixtures.

Water is the primary agent of both creation and deterioration of concrete. It is a necessary ingredient to produce concrete as water is needed in the cement hydration reactions to form the cement paste matrix which binds the constituents of concrete together. During the service life of concrete, when the hydration process, i.e., hardening of concrete, has continued for some time, on the contrary, water can play a degrading role as it enables many detrimental processes to occur. Penetration of water into concrete and transportation of aggressive agents by water are factors that initiate physical and chemical deterioration processes, and gradually cause de- crease in the durability of concrete. Oftentimes the physical and chemical processes of deterioration can act in a synergistic manner, further amplifying the degradation impact of each process. Examples of physical and chemical degrading processes are corrosion of reinforcing steel, freeze-thaw damages, sulfate attack and alkali-silica reaction (e.g. [1–3]).

The durability of concrete against physical and chemical degradation is largely dictated by its ability to impede moisture (and gases) from entering into and moving through the concrete, and that is mostly controlled by the mass transport properties of concrete. Moreover, the deterioration mechanisms typically depend on a crit- ical degree of saturation to initiate the degradation process. Hence, information on the internal moisture distribution and mass transport properties that govern the moisture transport and more generally mass transport would be essential in service- life prediction of concrete structures. Therefore, it is evident that development of monitoring techniques for moisture movement in cement-based materials has a significant role in deepening the understanding on mass transport mechanisms, and hence, finding ways to limit deterioration processes and improve durability of those materials.

In this thesis, the focus is on developing a moisture monitoring method based on electrical imaging for cement-based materials in controlled laboratory conditions, and hence, the discussion is mainly limited to laboratory testing methods. The aim is to study the capability of electrical capacitance tomography (ECT) for moisture

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monitoring applications and the following two research objectives are addressed 1. To study whether ECT can be used for imaging the evolution of moisture

during an ingress of water within cement-based materials.

2. To investigate whether ECT reconstructions could be further used for characterizing cement-based material’s moisture transport properties.

These aims are elaborated in Sections 1.1 and 1.2.

The primary water transport mechanisms in cement-based materials are governed by capillary absorption and permeability. Permeability controls moisture movement in saturated state and is driven by pressure gradient. In dry or partly saturated state (i.e. unsaturated state), the movement and ingress rate of water is largely controlled by capillary absorption through the action of capillary forces.

[2, 4–10] The majority of cement-based materials used in service are mostly in unsaturated state [3, 9], and hence, the water transport is dominated by capillary absorption. The unsaturated conditions are targeted also in this thesis.

In ECT, electrical capacitances are measured between electrodes placed around the perimeter of the target to reconstruct its internal electrical permittivity distribution. The permittivity distribution, in turn, is directly related to the moisture distribution. The relative permittivities of dry and hardened cementitious materials typically range from 3 to 9 [1, 2, 11–15] whereas the relative permittivity of water is approximately 80. Thus, the permittivities of cement-based materials vary strongly as a function of water content. Capacitive measurements have previously been utilized to assess the cover-zone moisture content in concrete [16, 17], however, the use of capacitive measurements for tomographic imaging of cement-based materials’ moisture conditions has not been reported until the publications of this thesis.

ECT has previously been applied to other porous media, for example, for estimating moisture distribution in soil [18, 19] and imaging the mixing and drying process of wet granules [20, 21].

Reconstructing the permittivity distribution is an ill-posed inverse problem [22, 23]. In practice, this means that the solutions of the inverse problem (permittivity reconstructions) are extremely sensitive even to small measurement and modeling errors. Hence, the reconstruction of the internal permittivity distribution requires mathematical modeling of the measurement process and proper computational methods associated with additional information of the target permittivity distribution. A brief review of ECT is given in Chapter 2. The discussion covers typical measurement setups, the mathematical model for the measurements and the reconstruction methods used in this thesis.

1.1 IMAGING OF UNSATURATED MOISTURE FLOWS WITH ECT A common and fairly straightforward method to monitor unsaturated water ingress in cement-based materials is the gravimetric method (ASTM C1585) where the total water uptake is measured by weighing the specimen at multiple times during the water absorption. Gravimetric method, however, does not give any information on the spatial moisture distribution within the specimen. A simple but labor intensive procedure to approximate the internal moisture profile is to slice the specimen at a sequence of times and to measure the moisture content of each slice by drying [3, 24, 25]. This approach, in addition to being destructive, needs several 2

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specimens to determine moisture distributions after different absorption times and has only moderate precision especially if water is required to reduce heat gener- ation when slicing the specimens. Due to unsuitability for long-term monitoring and laboriousness of the described monitoring approach, the use of non-destructive methods for moisture monitoring is highly preferable.

Many physical properties (electrical, radiation attenuation, wave velocity etc.) of cement-based materials are altered by the presence of moisture within them, and hence, these changing properties can be utilized as a basis of non-destructive methods for the assessment of the moisture flow [3]. For example, an electrical method, electrical impedance spectroscopy (EIS), has been used for monitoring flow of moisture based on alternative current (AC) impedance measurements between the electrodes that are embedded within the specimen at different locations [26–33].

This method, and a few other examples, such as capacimetry [16, 17, 34] and ground penetrating radar (GPR) [1, 2, 16, 17] can provide some approximate information about the spatial moisture distribution, for example, an approximate depth of the highly saturated region but no detailed information on how moisture is distributed.

The most accurate techniques to monitor water movement and moisture distribution within cement-based materials are tomographic imaging modalities. Ex- amples of such methods include nuclear magnetic resonance (NMR) (or magnetic resonance imaging (MRI)) [35–43], gamma-ray [34, 44, 45], X-ray [34, 46–51] and neutron imaging [52–60]. Quite recently, also an electrical imaging modality, referred to as electrical resistance tomography (ERT), that reconstructs the internal resistivity (or conductivity) distribution of the target has been successfully applied for imaging moisture distributions in cement-based and other building materials [16, 61–69].

Each of these tomographic methods has proven its ability to visualize and estimate the spatial moisture distribution of cement-based materials, but they also have some disadvantages. While NMR, neutron imaging and methods based on attenuation of electromagnetic radiation (gamma-ray and X-ray imaging) have a high spatial resolution, most of them are limited to small (thin) specimens due to high attenuation and/or are hardly available because of the high costs of the required measurement facilities and operational use. ERT has lower spatial resolution than the other aforementioned methods, but the costs of the measurement equipment are orders of magnitude smaller and there are no restrictions for specimen size. However, ERT can be properly applied only for imaging surface wet (electrically conductive) specimens or specimens that have relatively high moisture content since ERT measurements are based on electric current injections which require a good ohmic contact between the electrodes and target surface to be successfully performed [70, 71]. The main motivation of this thesis is to overcome the problem of using electrical imaging for electrically resistive (surface dry) specimens by using another electrical imaging technique, ECT, in those conditions. ECT is very similar imaging modality to ERT in many ways such as in spatial resolution, and it also has a mathematically challenging reconstruction problem. However, contrary to ERT, the (capacitive) measurements of ECT do not require ohmic contact between electrodes and the target, and they can even be performed without a physical contact with the target.

In this thesis, the applicability of ECT for imaging moisture flows inside cement- based materials was tested and in the experiments, mortar specimens (like concrete but without coarse aggregates) were used. In PublicationI, the main objective was to test whether ECT could detect moisture content changes within cement-based specimens during water absorption. The second aim of the first feasibility study was

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to investigate the ability of ECT to distinguish different water ingress rates between specimens with different moisture transport properties. In PublicationII, ECT was tested for imaging 2D water ingress within cement-based specimens. Especially the aim was to show the moisture movement and localize the moisture front. In the experiments, specimens with discrete cracks were also tested. The reconstructed 2D ECT images of the moisture flow were compared with photographs that provided approximate information on the moisture distribution of the specimens. In Publi- cation III, ECT was applied and tested for imaging fully 3D moisture movement within cement-based specimens. The 3D ECT reconstructions of the unsaturated moisture ingress were compared with numerically modeled evolutions of the moisture content that were based on finite element approximations of an unsaturated moisture flow model. The results obtained in Publications I-III are reviewed and discussed in Chapter 3.

1.2 CHARACTERIZATION OF CEMENT-BASED MATERIAL’S MOIS- TURE TRANSPORT PROPERTIES USING ECT RECONSTRUC- TIONS

As stated above, the durability of cement-based materials is mostly governed by the materials’ ability to impede flow of water. Thus, knowledge of moisture transport properties of the materials that characterize their tendency to take in and transport water would be of great importance when predicting the durability of those materials. Moreover, numerical moisture flow models and service life prediction modeling need some moisture transport properties as input parameters. One key transport parameter for cement-based materials is the water permeability, or the saturated hydraulic conductivity (SHC), which depends on the porosity, pore size distribution and pore connectivity. SHC is defined as the property that governs the rate of water flow into a porous medium. There exist well established methods originated from soil science to measure it, however, the measurement of SHC for cement-based materials is often very challenging and time consuming because of the low porosity and highly tortuous pore structure.

Tomographic imaging, apart from monitoring the internal moisture distribution, can also offer a tool for quantifying some moisture transport properties of cement- based materials. If transport properties of a material are spatially fairly invariant, by monitoring an unidirectional moisture ingress within a material using a tomographic modality, some properties can be estimated from the reconstructed evolutions of (1D) moisture profiles. One procedure is to apply Boltzmann transforma- tion to the reconstructed moisture profiles of different times to form a single master curve, from which, e.g., hydraulic diffusivity can be calculated [72, 73]. This approach has been used for estimating the transport properties of porous materials, for example, based on X-ray [34, 74], neutron imaging [75, 76] and NMR [35, 39, 77].

A slightly different procedure is to compare the reconstructed moisture profiles to predicted moisture profiles given some flow model and estimate the transport parameters (inputs of the flow model) by means of inverse analysis. This approach was utilized, for example, in [42, 43] where MRI was employed to estimate SHC of white mortar specimens.

PublicationIVinvestigates the further use of ECT reconstructions of the moisture flow process for characterizing material moisture transport properties. For estimation of the transport properties, the unidirectional flow process is approximated with 4

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a so-called sharp front (SF) model [3, 78], which assumes capillary absorption dominated 1D flow and homogeneous material properties. In the estimation scheme, specimens are imaged with ECT during water absorption, and the time-series of the reconstructed 1D permittivity profiles are used for tracking the approximate water front location (WFL) at discrete points in time. The WFL-data, in turn, is further utilized to estimate the SHC of the specimen using the SF model. The proposed SHC estimation scheme is tested numerically and experimentally. In the experimental part, the SHC estimates of mortar specimens are compared with SHC values measured using the falling head method (Darcy’s law). The ECT-based SHC estimation procedure and the results of the numerical and experimental studies are summarized in Chapter 4.

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2 Electrical Capacitance Tomography, ECT

Electrical Capacitance Tomography (ECT) is an imaging modality in which the spatially varying electrical permittivity distribution of a target is reconstructed based on measured electrical capacitances between electrodes placed around the target’s surface. ECT can be utilized for imaging targets/processes that feature contrast in electrical permittivity. The applications of ECT have mainly been in process industry, for example, imaging multi-phase flows [79–87], monitoring pneumatic conveyor [88–90] and fluidized bed [21, 88, 89, 91–94] systems, observing pharma- ceutical and chemical processes [95–97] and controlling mixing [20,91] and combus- tion [98–101] processes. In addition, ECT has been used for monitoring moisture conditions in soil [18, 19] and, in this thesis, for imaging moisture flows within cement-based materials.

The advantages of ECT are non-invasive and non-ionizing measurements, high temporal resolution, and low-cost measurement equipment compared to other imaging techniques such as MRI, neutron or X-ray imaging. The spatial resolution of ECT is, however, relatively low due to the ill-conditioned nature of the reconstruction problem [22, 23]. This means that the solution of the reconstruction problem is extremely sensitive to modeling errors and measurement noise. Hence, for reconstructing the target’s internal permittivity distribution based on indirect capacitance measurements, mathematical modeling of the measurement process and advanced computational methods are required.

In the following sections, practical aspects of ECT, mathematical model for ECT measurements and image reconstruction methods used in this thesis are reviewed.

2.1 MEASUREMENT MODEL

Figure 2.1 (a) illustrates an ECT sensor setup. The sensor consist of electrodes and the electrically grounded outer screen. The outer screen reduces the interference of external electromagnetic fields and is typically separated with an insulating layer (air or some other insulator) from the interior. Additionally, small grounded screen electrodes are normally used between the measuring electrodes. The ECT sensor is applied around the target. The electrodes do not necessary have to be in contact with the target, meaning that ECT is suitable for contact-free imaging. The one electrode layer setup presented in Figure 2.1 (a) is a setup for 2D permittivity imaging. For 3D ECT imaging, multiple layers of electrodes at different heights should be used to obtain information about the permittivity variations in vertical direction. Figures 2.1 (b)-(d) show a schematic illustration and two photographs of a 3D ECT sensor setup with 24 electrodes used in this thesis work. For more discussion and information on ECT sensors and setups, see e.g. [102–106].

There are numerous different measurement protocols in ECT. In the most common protocol, one electrode at a time is excited to some fixed electric potential, while the other electrodes are grounded, and electrical capacitances between the excited electrode and grounded electrodes are measured. After capacitances between all possible electrode pairs are measured, the excitation electrode is changed; with M

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Figure 2.1: (a) A schematic 2D ECT sensor setup, (b) an illustration of 3D ECT sensor (with 24 electrodes) used in this thesis, grounded external screen and screen electrodes are not shown here, (c)-(d) pictures of a real 3D ECT sensor.

measuring electrodes ^M(M₂⁻¹⁾ independent capacitance measurements are obtained as reciprocal capacitance measurements are excluded since capacitance between two electrodes is independent of the direction of the measurement. A few other excitation protocols for the ECT measurements are described, e.g. in [107, 108] where multiple electrodes are excited simultaneously. Although the measured quantity is often capacitance, the measurements can also be formulated in terms of electric charge with the relation

Q˜ =V^˜C˜ (2.1)

where ˜Qis the electric charge, ˜Vis the excitation potential and ˜Cis the capacitance.

Electric charges are more straightforward to formulate in the mathematical model than capacitances, especially in cases where multiple electrodes are excited simultaneously. In the ECT model that is described in the next section the measurements are formulated as electric charges.

2.1.1 Forward model

The derivation of the mathematical model (i.e. forward model) for the ECT measurements is based on the Maxwell’s equations of electromagnetism

∇ ·^D=ρ_f (2.2)

∇ ×^E=−^∂B

∂t (2.3)

∇ ·^B=0 (2.4)

∇ ×^H=j_f+^∂D

∂t . (2.5)

Equations (2.2)-(2.5) are, respectively, Gauss’s law for the electric field, Faraday’s law of induction, Gauss’s law for the magnetic field, and Ampere’s circuital law in differential form. Dis the electric displacement field (C/m²),ρ_f is the free charge density (C/m³),Eis the electric field (V/m),Bis the magnetic flux density (T),His the magnetic field (A/m), and j_f is the free current density (A/m²).

The medium is assumed to be linear and isotropic, and hence the relationD=εE can be used where ε is the electrical permittivity. Moreover, the so-called quasi- 8

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electrostatic approximation is made. That is, although alternative current (AC) potential excitations are used, the time dependence is neglected (time derivatives are zero) and only the amplitudes of the quantities (D, E, etc.) are used. The quasi- static approximation is a valid assumption considering that the AC frequency of the potential excitation used in the ECT measurements is typically in MHz range and it corresponds to wavelengths much larger than the dimensions of the ECT sensor. The quasi-electrostatic approximation leads to∇ ×^E= 0 which means thatE field is circulation free, and it can be expressed with a scalar potential functionuas E= −∇u. Using relationsD= εEand E=−∇u, Gauss’s law (2.2) and assuming there are no free charges in the medium (ρ_f =0), yields to Poisson equation

∇ ·(ε(x)∇^u(x)) =0, x∈^Ω^. ^(2.6) In Equation (2.6), x is a three (or two) dimensional position vector inside the computational domain Ω, and u(x) is the electric potential. The electrical permittivity can be written as ε(x) = ε0εr(x), where ε0 is the permittivity of free space (ε₀≈8.8542·10⁻¹²Fm⁻¹), andεr(x)≥1 is the relative permittivity of the medium.

The forward model for the ECT measurements used in this thesis consist of the partial differential equation (PDE) (2.6) and the following boundary conditions

u(x) +ζε(x)^∂u(x)

∂nˆ =U, x∈^∂Ω_el⁽⁾^,=1, ...,N_el (2.7)

−

∂Ω⁽⁾_el ε(x)^∂u(x)

∂nˆ dS=q, =1, ...,N_el (2.8)

∂u(x)

∂nˆ =0, x∈^∂Ω\^∂Ωel, (2.9) where N_el is the total number of electrodes including both the measurement and grounded screening electrodes, U is the applied potential (excitation or zero) on th electrode, ζ is a coefficient related to contact impedance of the th electrode, causing a small potential drop on the electrode surfaces, q is the electric charge on th electrode, ˆn is the unit normal vector pointing outward from the electrode surface or the boundary of the domain, and ∂Ω_el = ₌₁^N^el ∂Ω⁽⁾_el is the union of patches covered by the electrodes.

The boundary conditions (2.7)-(2.9) can be interpreted as follows: The condition (2.7) assigns the applied constant potential on the excitation and grounded electrode(s) during one excitation cycle. The coefficients ζ in (2.7) denote contact impedances between the electrodes and target. In this thesis, parametersζare fixed to be a small constant because based on simulation and experimental tests in volt- age excitation measurement technique, the drop in electric potential due to contact impedances is very small, and the solutions of the boundary value problem (2.6-2.9) are not particularly sensitive to the choice of this parameter. The condition (2.8) describes the induced electric charges on the electrodes due to the potential excitation. The last condition is valid at the electrode-free part of the domain boundary, and it means that the electric field is zero in the direction of the unit normal at the boundary. In addition to conditions (2.7)-(2.9), it is also required that the charge conservation law

N_el

=1

∑

q=0 (2.10)

is obeyed.

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The forward model presented above differs slightly from the conventional PDE model for ECT measurements. In the conventional model (see e.g. [20, 21, 106, 109–

111]), the electric potential distribution is first solved separately for each electrode excitation, and then the condition (2.8) is employed afterwards to compute the electric charges on the electrodes. This requires a numerical integration over potential gradient, which can be numerically inaccurate if sufficiently dense discretization is not used. An adequate discretization makes the integration accurate but at the same time it increases the computational burden.

In this thesis, the ECT forward model (2.6-2.10) presented above is based on com- plete electrode model (CEM) [112–114] of electrical impedance tomography (EIT) with the assumption that electrical conductivity, σ(x), of the target is neglected (σ = 0). In contrast to the conventional ECT model, using the model (2.6)-(2.10) completely avoids the numerical integration over the potential gradient.

2.1.2 Finite element approximation of the forward model

The forward problem in ECT is to solve the electric potentialu(x)inside the domain Ω and the electric charges on the electrodesq = q₁,q2, . . . ,qN_el

(the measurable quantity in ECT), when the permittivity distribution and the excitation potentials are known. To solve the forward problem, one has to employ numerical methods since analytical solution of the forward model (2.6)-(2.10) is known only in extremely simplified situations due to the complicated boundary conditions. In this section, an approximative solution for the forward model using finite element method (FEM) is shortly reviewed. The derivation of the variational form and more detailed FE -approximation of the ECT forward model is presented in Appendix A. For more general theory of FEM, see [115].

In FEM, the computational domain Ω is divided into a mesh of finite number of elements that are connected through nodes. A finite dimensional approximation of the solution is sought in this mesh for the unknown functions. In the case of ECT, approximations for the electric chargesq≈^qh, electric potentialu(x)≈^uh(x) and permittivity distributionε(x)≈^εh(x)are written to solve the forward problem numerically. The permittivity distribution is approximated as

ε(x)≈^εh(x) =

∑

N i=1

εiϕi(x), x∈^Ω ^(2.11) where Nis the number of nodes in the finite element mesh. In this thesis, the basis functions ϕ_i(x) are chosen piecewise linear (first order polynomials) such that the coefficients ε_i represent permittivity values in the nodal points. In the following, the finite dimensional representation of the permittivity distribution is denoted as ε = (ε₁, ...,ε_N)^T ∈ ^R^N. For the unknown electric charges q and electric potential u(x), the following approximations are chosen

q≈^qh=

N_el−1 k=1

∑

˜

q_kn_k, n_k∈^R^N^el (2.12)

u(x)≈^uh(x) =

∑

N j=1

u_jφ_j(x), x∈^Ω^, ^(2.13) where nk are constant vectors such that the condition (2.10) is fulfilled, e.g., nk = (1, 0, ...,−1, 0, ..., 0)^T, where the(k+1)th component is -1 and, furthermore, ˜qk are 10

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the associated coefficients. The basis functionsφj(x)are again chosen as piecewise linear, and hence the coefficients uj represent the finite element solution u_h in the nodal points.

The variational formulation of the forward model (2.6)-(2.10) and approximations (2.11)-(2.13) result in a system of linear equations that can be written as a matrix equation (see Appendix A)

Ay= f, (2.14)

where A is the stiffness matrix and f is the data vector. y^T = (u1, ...,uN, ˜q1, ...,-

˜

qN_el₋₁)^Tis the solution vector that holds the coefficients of the approximations (2.13) and (2.12) to be solved.

By solving Equation (2.14) asy = A⁻¹f, coefficients of the approximative solution for the forward problem are obtained. The electric chargesqcorresponding to one electrode excitation can be approximated using Equation (2.12)

q=

N_el−1 k=1

∑

˜

q_kn_k =M^y^, ^(2.15)

whereq= (q₁, ...,q_N_el)^T,yis the solution of Equation (2.14) and the matrixM^is M=0^¯ n₁ n₂ . . . n_N_el₋₁

∈^R^(N^el^)×(N+N^el⁻¹⁾^, ^(2.16) wheren_kare the basis vectors and ¯0is a zero matrix of sizeN_el×^N.

As already discussed, the actual ECT data is measured using multiple electrode excitations, and for each excitation (U⁽ⁱ⁾) the forward problem needs to be solved individually. Assuming that the actual ECT capacitance measurements C ^{are con-} taminated with additive noiseνand converted to electric chargesQusing Equation (2.1), the observation model for the ECT measurements can be written in the form



 Q⁽¹⁾

... Q^(p)



=





h(ε,U⁽¹⁾) ... h(ε,U^(p))



+



 ν1

... νp



 (2.17)

where Q^(j) includes the measured electric charges and h(ε,U^(j)) = M^y(ε,U^(j)), i.e., forward solution (2.15), corresponding tojth excitationU^(j). Equation (2.17) can be expressed more shortly as

Q=H(ε) +ν, (2.18)

where Qcorresponds to a full set of ECT electric charge data measured for p electrode excitations, H(ε) includes the solutions of the forward problem for the corresponding excitations andν = (ν₁^T, ...,ν_p^T)^T. The observation model (2.18) is non- linear with respect to the permittivity distribution. Note that the data corresponding to the reciprocal measurements should be excluded from the computed ECT data if the actual measurement data does not include those measurements. In addition, if grounded screen electrodes are modeled in the forward model, the computed measurements corresponding to those screen electrodes need to be excluded as well.

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2.2 RECONSTRUCTION OF THE PERMITTIVITY DISTRIBUTION This section reviews the methods used in this thesis to solve the inverse problem in ECT, i.e., how to reconstruct the internal permittivity distribution based on noisy electric charge/capacitance measurements.

ECT belongs to the class of diffusetomography modalities where the aim is to estimate the spatially varying coefficients of a PDE based on noisy measurements obtained from the boundary. Because of the diffusiveness, each measurement is dependent on all of the permittivity within the target, resulting in measurements that are sensitive to large-scale permittivity variations and insensitive to fine-scale changes in the permittivity. As a consequence, the image reconstruction problem is more challenging than, for example, inhard-fieldtomographies such as X-ray tomography, where each measurement carries information from a very restricted region of the target. Diffuse tomographies have typically a relatively low spatial resolution, because the image reconstruction problem has a nature of anill-posed inverse problem.

The ill-posedness of the inverse problem is associated with the characteristics of the forward mapping. The forward mapping of ECTH(ε)is such that a relatively large change inεcauses only minor changes inH(ε)(computed observations). In other words, a small change in the measurements corresponds to a large change in the permittivity estimate. This practically means that in the presence of measurement noise and modeling errors, classical solutions (such as the least squares (LS) solution, arg minε

^Q−H(ε)²) are typically non-unique and unstable. As a result, in order to solve the reconstruction problem of ECT, in addition to accurate modeling of the measurements, the solution requires incorporating some regularization (in deterministic framework [116]) or additionala prioriinformation about the target permittivity (in Bayesian statistical framework [22]).

In this thesis, the inverse problem of ECT is solved based on the observation model (2.18) and the reconstruction problem is treated in the deterministic framework as an LS-problem augmented with a regularizing penalty functional. The regularizing functional is constructed such that it favors expected features of the target permittivity distribution and penalizes for improbable features. For instance, the expected features may be smooth or, alternatively, sharp-edged permittivity variations on a homogeneous background. Here, it is noted that in the Bayesian framework of inverse problems, the permittivity estimates of this thesis can be interpreted as Maximum a posteriori (MAP) estimates, and the regularizing penalty functional as a prior potential function related to a statistical model of the permittivity. In this thesis, the discussion of the inverse problem is limited to deterministic framework, yet Bayesian interpretation is utilized in the selection of the parameters in the regularization functional. For a comprehensive information on solely using the Bayesian statistical approach for solving the inverse problem in electrical tomographies, see, for example, [22, 110, 117, 118]. For a collection of different reconstruction methods used in ECT, see [104, 106, 111, 119–125].

In this thesis, two different imaging schemes,differenceandabsoluteimaging, are used. In the former scheme, the change in permittivity between two time instants is reconstructed on the basis of the difference in the data sets. In absolute imaging, the absolute permittivity values are estimated using a single set of ECT measurements.

These two imaging schemes are elaborated in the following two subsections. The selection and construction of the regularization functional used in this thesis are described in Section 2.2.3. Finally, the effect of the electrical conductivity on ECT measurements and reconstructions is briefly discussed in Section 2.2.4.

12

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2.2.1 Difference imaging

In difference imaging, the temporal change in the permittivity is estimated on the basis of the difference in the ECT data corresponding to two time instants. Consider two ECT measurementsQ₁andQ₂corresponding to permittivitiesε₁andε₂at time instants t₁ and t₂, respectively. The observation models corresponding to the two ECT measurements can be written as

Q₁=H(ε₁) +ν₁ (2.19)

Q₂=H(^ε2) +ν₂, (2.20)

where the measurement noises ν_i are assumed to be mutually independent and equally distributed Gaussian random variables with mean ν^∗ and covariance Γν, i.e.,ν_i ∼ N(ν^∗,Γν).

In linear difference imaging, the non-linear forward mappingH(ε)is linearized at pointε_linusing the Taylor series expansion. By omitting the higher order derivatives, the forward mapping is approximated as

H(ε)≈H(ε_lin) +J(ε_lin)(ε−^εlin), (2.21) where J(ε_lin) = ^∂H(ε_∂ε^lin⁾ is the first derivative of the forward mapping, i.e., the Jacobian matrix and it is evaluated at the linearization point. The computation of the Jacobian matrix is derived in Appendix A. Substituting the approximation (2.21) to (2.19) and (2.20), the following expression for the linearized observation models is obtained

Q_i =H(^εlin) +J(ε_i−^εlin) +ν_i, i=1, 2 . (2.22) Using the linearized models and subtractingQ₁fromQ₂, an observation model for the difference data∆Q=Q₂−^Q1can be written as

∆Q= J∆ε+∆ν, (2.23)

where∆ε=ε2−^ε1is the permittivity change and∆ν=ν2−^ν1is the difference in the noise terms.

Given the model (2.23) and considering the estimation of∆ε as an LS-problem added with a regularization term, an estimate for∆εis obtained as a solution of the minimization problem

∆ε=arg min

∆ε

^L∆ν(∆Q−^J∆ε)²+p∆ε(∆ε) , (2.24)

where L_∆ν is the weighting matrix defined as L^T_∆νL_∆ν = Γ⁻_∆ν¹, where Γ_∆ν, the covariance of the measurement noise term ∆ν is Γ_∆ν = Γν₁+Γν₂ = 2Γν. The minimization problem can be equipped with an additional non-negativity constraint for the permittivity change (∆ε ≥ 0) (this was done in Publications IIand III). In those publications, the non-negativity constraint was implemented using quadratic barrier function B(∆ε) [126]. Lastly, p_∆ε(∆ε) is the regularization functional that contains the prior information on the unknown∆εdistribution.

The selection and construction of the regularization functionalpα(α)for the unknown parameter will be described in more detail in Section 2.2.3. In this thesis, however, the regularization functional is always chosen to be of the quadratic form Lα(α−^α^∗)² where α is the unknown, α^∗ is the expected value of the unknown

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and Lα is the regularization matrix. In such a case and if the non-negativity constraint is not written, the minimization problem (2.24) is linear w.r.t. ∆ε and the solution is computed explicitly with one step as a generalized Tikhonov regularized solution [127]

∆ε =J^TΓ⁻¹_∆νJ+W_∆ε₋1

J^TΓ⁻¹_∆ν∆Q+W_∆ε∆ε^∗

, (2.25)

where W_∆ε = L^T_∆εL_∆ε and L_∆ε is the regularization matrix. If the non-negativity constraint is included, the minimization problem (2.24) is non-linear w.r.t. ∆εand is solved iteratively using Gauss-Newton (GN) method [126–128]. In this case, with the initial value∆ε⁽⁰⁾, the iteration step∆ε⁽ⁱ⁺¹⁾,i≥0 is computed as

∆ε⁽ⁱ⁺¹⁾=∆ε⁽ⁱ⁾+β_id⁽ⁱ⁾_∆ε, (2.26) where the parameter βi controls the step size in the search direction d⁽ⁱ⁾_∆ε which in turn is given by

d⁽ⁱ⁾_∆ε =

J^TΓ⁻_∆ν¹J+W_∆ε+B(∆ε⁽ⁱ⁾) ₋1

·

J^TΓ⁻_∆ν¹(∆Q−J∆ε⁽ⁱ⁾)−W_∆ε(∆ε⁽ⁱ⁾−^∆ε^∗)−B(∆ε⁽ⁱ⁾)

, (2.27) whereB(∆ε)andB(∆ε)are the Hessian matrix and gradient of the barrier function B(∆ε), respectively.

In this thesis, the ECT data consist of sequential measurements during the flow of moisture and the aim is to reconstruct the temporal changes in permittivity to keep track how moisture is propagating within the material. The initial dry state permittivity (at time before water absorption is initiated) is approximated as spatially constant permittivity and it is estimated by doing a one-parameter LS fit to the data measured before initiating the water absorption. The performance of linear difference imaging depends on the selection of the linearization point ε_lin. In this thesis, the linearization point is selected to be the permittivity (distribution) corresponding to time instantt1, i.e.,ε₁.

Two different difference imaging approaches were used in this thesis to reconstruct the temporal evolution of the permittivity during moisture flow: 1) the (standard) globally linearized (GL) difference imaging and 2) the sequentially linearized (SL) [129] difference imaging. Using the notations from above, the procedures of these two methods to estimate the temporally changing permittivity is based on the following choices

GL :ε_lin =ε₁=ε⁽⁰⁾, ∆Q=Q^(k)−^Q⁽⁰⁾^, J= J(ε_lin), ε^(k)=ε⁽⁰⁾+∆ε

SL :ε_lin =ε₁=ε^(k⁻¹⁾, ∆Q=Q^(k)−^Q^(k⁻¹⁾^, J= J(ε_lin), ε^(k)=ε^(k⁻¹⁾+∆ε,

where the upper indexes (0,1,2,...,k-1,k) inside brackets correspond to different measurement times. In the GL method, the linearization point is fixed to be the state 14

Imaging moisture flows in cement-based materials using electrical capacitance tomography

uef.fi

Dissertations in Forestry and Natural Sciences

ANTTI VOSS

IMAGING MOISTURE FLOWS IN CEMENT-BASED MATERIALS USING ELECTRICAL CAPACITANCE TOMOGRAPHY

ANTTI VOSS

Antti Voss

IMAGING MOISTURE FLOWS IN CEMENT-BASED MATERIALS USING

ELECTRICAL CAPACITANCE TOMOGRAPHY

TABLE OF CONTENTS

1 Introduction

2 Electrical Capacitance Tomography, ECT

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