Modeling pore continuity and durability of cementitious slurries under high temperatures Текст научной статьи по специальности «Строительство и архитектура»

Строительные материалы и изделия / Building Materials and Products

Koenders E.A.B.

EDUARDUS A. B. KOENDERS, Ph.D., Professor Construction and Building Materials, Department of Civil Engineering and Geodesy, Technical University Darmstadt. Rm 209, Franziska-Braun-StraGe, Darmstadt, 64287, Darmstadt, Germany, e-mail: [email protected]

Modeling pore continuity and durability of cementitious slurries under high temperatures

In this paper, results of a numerical study on pore continuity, permeability and durability of cementitious slurries for carbon sequestration projects are presented. The hydration model Hymostruc is used to simulate and visualize 3D virtual microstructures which are used to demonstrate the contribution of capillary pores to the continuity of the capillary pore system embedded in an evolving cementitious microstructure. Once capillary pores are blocked due to ongoing hydration, transport of CO2 species through the microstructure is avoided which may protect the slurry from leakage. Evaluating the pore continuity of the capillary pore system during hydration of the microstructure is therefore indispensable for a robust cementitious sealing material and is the main objective for slurry design. Simulations are conducted on slurries exposed to ambient temperatures of 20 °C, 40 °C and 60 °C, and a durability outlook regarding the CO2 ingress is given as well. Aggregates and associated interfacial transition zones are introduced in the slurry system that may cause alternative porous path ways through the system. Pore continuity analysis show the relevance of numerical simulations for assessing the capillary pore structure inside an evolving microstructure in relation to its sealing and durability performance.

Key words: concrete, cement, pore, microstructure, permeability, durability.

Introduction

The slowly but still more clearly emerging change of our climate, is enforcing the construction sector to progressively embracing a more sustainable use of our natural resources and energy conservation while minimizing the emission of CO2 into the atmosphere [28, 29, 39]. Capturing and storing CO2 in underground layers or exhausted oil and gas reservoirs is considered one of the solutions to re-stabilize the CO2 concentrations in the atmosphere through a reduction of the emitted amounts of CO2 to acceptable levels [31, 39]. Exhausted oil and/or gas reservoirs can potentially be used as storage facilities for the worlds CO2 "over capacity" [2, 6, 43]. These massive reservoirs of porous (permeable) rock situated in deep underground layers were depleted by exploration activities using conventional wells, drilled with the main purpose to extract the natural resources such as gas or oil for an often protracted duration [9]. Reusing these potential storage capacities for CO2 sequestration may also require newly drilled CO2 injection wells, which should be designed in order to withstand the new local conditions, i.e. CO2 at high temperature and high pressure [2, 14]. Designing new wells to avoid potential gas leakage, but also monitoring the structural integrity and sealing ability of the existing casing cement slurry that embeds abandoned wells is one of the key issues to be

© Koenders E.A.B., 2015

considered when deciding upon reusing depleted underground reservoirs for CO2 storage [8, 10]. Besides, relevant issues that may contribute to the loss of integrity of the well, i.e. causing gas leakages in the existing cement slurry, are closely related to those actions that may introduce (micro-) cracks and affect the pore discontinuity causing alternative pathways for the CO2 to escape 8. Interfaces between in-place cement slurry and rock formations, migration of the CO2-rich phase (aqueous, gas or liquid) through fractures or diffusion through the connected capillary pore space are considered to be most threatening mechanisms that may harm the cement slurries sealing capacity of a well, causing loss of structural integrity [8, 12, 25, 27, 44]. Arguments on the applicability of Portland cement as the most reasonable binder to plug CO2 wells are detailed in [27].

Carbonation degradation in cement-based materials is a coupled chemical equilibrium/diffusion phenomenon, that may also include fracture mechanics. Kinetics of transport-reaction processes are generally described by a simplified continuum (homogenized) model [36] where the actual evolution of the pore morphology are neglected. The models proposed are based on an apparent diffusivity including the reaction process, which can account explicitly for the reaction of CO2 with the cement-based phases [3, 33, 35]. These models run at the meso and macro scale level and use a homogenisation approach to upscale microstructural properties, whose dependences on the material and hydration parameters are considered by over-simplified empirical correlations (e.g. diffusivity-porosity correlations [38]). On the other hand, the DuCOM model [20] offers an integrated multi-scale computational platform where the microstructure and it's moisture state govern basic degradation processes. However, while the DuCOM model simulates the microstructure development by assuming a mono sized particle dispersion, this paper is based on Hymostruc model that simulates growth of poly-dispersed particles and includes the effect of addition of pozzolanic supplementary materials. Microstructure properties are then directly evaluated from the 3D virtual microstructure of the hydrating cement-based matrix. In this paper, a flow model at micro-scale level is currently being implemented enabling the possibility to simulated 3D flow of harmful species through an evolving microstructure while accounting for a change of the pore structure, and with this the permeability. In a first approach, the model accounts for evolution of the pore morphology during hydration at a water saturated state, but neglects carbonation reactions. Moreover, the paper addresses the effect of adding aggregates or non-reactive fillers to a cement-based slurry that introduces a potential alternative route for moisture or dissolved matter to move through a cement-based system. These zones of increased porosity around aggregates may, once inter-connected, influence the permeability, and with this, affect the sealing performance of a cement-based slurry system significantly. Results of a numerical study will be reported with emphasis on the potential leakage of CO2-rich phase through the capillary pore space of an evolving cement slurry microstructure that seals-off the casing in a wellbore. After fresh slurry placement and final set is reached, early hydration commences and silicate reactions will start to form calcium silicate hydrates which are considered the main building blocks of a microstructure [42]. Calcium hydroxide is formed as well. Both hydration products are susceptible to reacting with the stored CO2. These reactions will increase the capillary porosity and change the pore-structure of the cement slurry from the interior [26, 27]. The damage caused by these processes may weaken the microstructure and increase the pore continuity of the capillary pores. Therefore, in order to scrutinize the mechanism described so far, a numerical study has been conducted and results are presented in this paper. In section 2, the background of the numerical model is described together with the formation of microstructure and capillary pore space. From this, in section 3, a modeling approach is discussed on how the pore continuity in a 3D virtual microstructure can be calculated. Section 4 reports the pore continuity development for simulated cement slurry at three different ambient temperatures, while in section 5 the potential threat of CO2 on the durability of cement slurries is discussed. The paper ends with section 6 showing the effect of the interfacial transition zone (ITZ) on the sealing performance of a cement slurry.

Numerical model for hydration

The Hymostruc model is 3D code that can be used to simulate the hydration evolution of a virtual microstructure of cementitious materials [5, 24, 37, 38]. The model is based on a 3D scheme and calculates the development of the microstructure as a function of the particle size distribution, water-cement ratio, chemical composition and mix temperature for Portland and blended cement mixtures. The processes that run during the hydration reaction can be distinguished into different categories including morphological, physical, chemical and thermo-dynamical. In view of the development of the cementitious microstructure, all these categories have their own particular characteristic and affect the hydration reaction in a certain way. In Hymostruc, these processes, together with the particle interactions that occur during hydration, are modeled explicitly. Within the model, Portland cement clinker is the primary reactive compound (Fig. 1a). However, recently the model has been extended and made suitable to simulate the hydration reactions of Portland cements blended with slag, silica fume or fillers (Fig. 1b). Detailed description on the implementation of the model that accounts for hydration of blended systems (cement and pozzolans) can be found in Internal report [24].

An envelope shape needs to be defined for the initial locations of the particles which then form the initial state of the microstructure. The envelope needs to comply with periodic boundary conditions of those particles that protrude an envelope plane. This approach enables the particle filling algorithm to accurately comply with the imposed water/cement ratio. Particles are stacked in the envelope shape based on random selection of a location with equal probability. At first, the largest particles are placed followed by the smaller once, according to the particle size distribution. After having placed all particles, the initial state of the 3D microstructure is ready to be used for calculating the hydration process. From the 3D particle configuration (see Fig. 1a), inter-particle spacing's and paste density can be calculated for each central particle in the system [5]. Based on this information, the distance between the hydrating particles is known which can be used for calculation of the particle growth. The hydration model offers two different possibilities for simulating the particle hydration as well as the inter-particle interactions, e.g. according to a statistical cell concept or according to a full 3D random particle concept. While the cell calculates the inter-particle distances as an average value from an elementary paste volume that contains volumes of water and (blended) cement grains up to a certain fraction, the full 3D concept calculates the inter-particle distances directly from the generated 3D particle structure. The cell concept, which is applied in this article, forms the basis for the potential outward growth approach of the hydrating particles.

a) Portland cement b) Blended cement

2

Fig. 1. Examples of a 0.45 wcr 3D virtual microstructure consisting of Portland cement (400 m /kg) and Portland

2

cement blended with 20% pozzolans (500 m2/kg), at the initial state

A cell is defined as a cubical elementary volume in which a central particle with diameter x is the largest particle. It further contains water and (pozzolanic) cement grains, all with a diameter smaller than the central particle x. The cell density px can be calculated from

(K<x + Vp<*) 3 3

-j [pm /pm ] (1)

x)

рх =

volume of binder in cell Ix

volume of cell Ix

V:

+V

c<x p<x) + Vc<x + Vp<xJ

where Vc<x and Vp<x is the volume of all cement and pozzolan particles with a particle diameter smaller or equal to the diameter of central particle x, respectively. Vw is the volume of water that complies with the water to cement and/or water to binder ratio, and Ix is the total cell volume corresponding to the total volume of the cement, pozzolan and water up to diameter x. The cell density can be used for calculating the shell density which is the expanded shell that surrounds the central particle x and is driven by the outward growth of the particle hydration. The average shell density psheii,x, representing the mass of blended (cement and pozzolanic) particles in a shell that surrounds the central particle, can be calculated according to

рshell ,x

volume of binder in spherical shell total shell volume

р shell ,x

рx

■ I- v..

or Ix vx (2)

where vx is the volume of the central particle x, which can either be cement or pozzolan (when simulating hydration of blended systems).

The reaction front is considered to grow inwards forming inner C-S-H gel or high density C-S-H gel, while in the outward direction an outer C-S-H gel or low density C-S-H gel is formed which causes the particle radius to expand. For the calculation of the actual expansion of the hydrating particles, the so-called particle expansion mechanism is applied (Fig. 2) [5, 24]. The expansion mechanism describes the

outer growth of a spherical central particle with diameter x, while accounting for the geometrical overlapping and embedding of the (partly) hydrating cement and/or pozzolanic particles situated in close vicinity. The initial outward growth is calculated from the mass balance between the reactants and the reaction products created, while taking into account the kinetics and stoichiometries of the different clinker minerals as well. Based on this, a volume ratio can be derived for the primary silicate reactions, uc, as well as for the secondary pozzolanic reactions, vf. Following this, a relation has been established between the chemical nature of the cementitious system and the potential expansion of grains in a developing microstructure, i.e. a

Pozzola

embedded particles

expansion caused by embedded particles

outer product

inner product

Fig. 2. Schematic impression of expansion mechanism [5]

relation between the type of cementitious material and the morphology of the microstructure being formed. The volumetric expansion of the individual hydrating particles Vou,ex,xj and Vou,ex,xj with diameter x at time j and an initial outer volume V'^ and Vouxj can be calculated by the following set of equations for cement (sup c) and pozzolan (sup p), respectively (eqs 3 and 4).

vc

V

ou;x, j

ou,ex ;x, j

i-Cix-s^j )-if «j )+f «j ))

(3)

c

Vp .

t/ p __ou;x, ]

ou,e " i-ax-spux,) ■{/ )+f«,)), (4)

where, Z is the density of the shell surrounding a central particle. The factors f(ac<xj) and f(ap<xj) account for the additional growth of the central particle due to the overlap with smaller particles according to:

f«.j ) = ^ -{i + («/ -1)} (5) f «J ) = VPx -{i + - 1) •<; }

(6)

The grains a <x,j and aP <Xj are the degree of hydration of the overlapped particles and v <x and V <x are their initial volumetric ratios (see also 523) . With these equations the expansion of the four cementitious Bogue phases (C3S, C2S, C3A and C4AF) and the pozzolanic material can be calculated to form an evolving virtual microstructure [5, 24] (Fig. 3).

2

Fig. 3. 3D virtual microstructure of wcr 0.45 hydrated slurry for a) Portland cement (400 m /kg) system (ac=0.84),

2

and b) Portland cement blended with 20% pozzolan (500 m /kg) system (ac=0.70, ap=0.93)

ВЕСТНИК ИНЖЕНЕРНОЙ ШКОЛЫ ДВФУ. 2015. № 1 (22) / FEFU: SCHOOL OF ENGINEERING BULLETIN. 2015. N 1/22

From the reaction kinetics of the individual cement minerals, i.e C3S, C2S, C3A, C4F and pozzolanic phases, the molar volumes per gram that dissolve with time can be calculated as well [26]. With this, the hydration model implicitly accounts for the effect that smaller particles hydrate faster than the bigger particles and show a much faster evolution of the degree of hydration. However, in terms of mass, the finer cement and pozzolanic fractions (silica fume, fly ash) contribute less. For blended cementitious systems the degree of hydration as is calculated according to

as(t) = a, = mc -a] + mp ■aj

(7)

where mc is the mass fraction of the cement phase and mp the mass fraction of the pozzolanic phase. The degree of hydration aCj of the cement phase is composed of the individual contributions of the four independent Bogue phases, i.e. C3S, C2S, C3A and C4AF, representing the phase fractions inside a cementitious clinker, according to (eqs 8 to 11)

C3S

¿<f =

S-C3S 1 c 3

r

x

C3S

1 -

OC 3S , i С"

C2S

AaXCf

C3A

А<Г =

C4AF

1 --

£

x,j-1

1 -

eC2S . л XC2S

oC 3 A 1 £x, 3-1 c 3

r

x

1 -

j + Л£

x,j

Ла

C 4 AF

1 -

£

C 4 AF x,j-1

1 -

£

C 4 AF x,j-1

+ Л£

C 4 AF

pore volume [cm3 / cm3] water filled pores a

---/Vw

V

por

V

wat

pore volume filled with water

0 1 2 _ 10 10 10 10 10

pore diameter [um]

Fig. 4. Schematic representation of the pore volume distribution as a function of the pore diameter [5, 23, 40]

(8)

(9)

(10)

(11)

For the pozzolanic phase a similar approach can be followed. The incremental change of the degree of hydrating pozzolanic particles is calculated according to (eq. 12): Pozzolanic

ЛаР, =

1 -

£

x,j-1

1 -

£pP,-1 +Л£

x,j

(12)

Ф

por

3

c

r

X

3

3

c

c

r

r

X

X

3

c

r

X

3

3

c

c

r

r

X

X

3

3

c

c

r

r

X

X

In these equations Sx is the actual progress of the reaction front into a particular grain, while ASx is the incremental increase of this reaction front. The degree of hydration of the system is calculated from the total amount of cementitious material that has reacted relative to the initial amount of cementitious materials present.

Integrated particle kinetics

In the hydration model Hymostruc, the formation of a 3D microstructure is simulated with the hydrating cement particles modeled as expanding spheres, eqs. (1-12), and follow an integrated kinetics approach according to eq. (13). In this equation K0 is the initial rate of hydration reaction of the individual cement or pozzolan particles, F1 is the Arrhenius function and the three Qi factors account for the state of water in the microstructure during hydration. The last part in the equation describes the change of the reaction process from phase boundary to diffusion controlled and also accounts for the effect of the reaction temperature F2 on the morphological outward growth of the C-S-H gel and, with this, on the capillary porosity. The reaction front of any phase into an individual reacting particle with diameter x, at time t can, therefore, be described with the following equation, called 'Basic Rate Equation' (BRE):

AS

-j = k0 .Ц-Q2 .Q3 • f •

1 A

S

(f2)a •1s-

(13)

In this rate equation A8j+1 represents the incremental increase of the penetration depth of the reaction front, developed in the upcoming incremental time step Atj+1, while Str and ¿x,j are the transition thickness and the thickness of the total C-S-H layer, respectively. The factor A accounts for the change of the particle reaction from phase boundary to a diffusion controlled process and pi and p2 are two empirical constants [5]. The first reduction factor Q1 refers to the hydration of the embedded but still incompletely hydrated cement and/or pozzolanic particles that are captured by the expanding outer shell of either a hydrating cement or pozzolanic particle. The still incompletely hydrated cement and/or pozzolanic particles embedded in the shell (Fig. 2) may consume a certain amount of the available water needed for further hydration of the central cement or pozzolan particle. This means that the rate of hydration of the central cement or pozzolanic particle will be affected by the presence of embedded and still incomplete hydrated cement and/or pozzolanic particles as well. It also means that the kinetics of an individual particle depends on its particular position in a microstructure, which is determined by its so-called "neighbor" particles. It makes the rate of hydration a unique property for each individually hydrating particle in the system. In the cell concept as presented in this paper, the kinetics mechanism is considered per fraction. Therefore, the reduction factor Q1 that affects the rate of the central cement particle x holds for all particles in a fraction and can be described according to the following formulation

~ Awx i

a = ■ x,j

Aw + m • Awc + m -Awp ^

x, j c em;x,j p em;x,j (14)

where AwXj, is the incremental water consumption of the central particle x (either cement or pozzolan) and Awcem;xj and Awpem;Xj, are the incremental water consumptions of the embedded cement particles and/or the embedded pozzolanic particles, respectively (see Fig. 2). Furthermore, mc and mp are the mass fractions of the embedded cement and pozzolanic particles of all particles with a diameter smaller than the central particle with diameter x. The second reduction factor Q2 refers to the water shortage in the capillary pore system [5] and its associated effect on the partial emptying of the pores and, consequently, locally ceasing of the hydration process because of that [5, 23]. When considering this concept in view of the addition of a pozzolanic phase, the microstructural parameters that are affected by the hydration of the cement and

я

pozzolanic phase are both the actual volume of capillary water Vcap (due to the primary and secondary pozzolanic reaction) and the actual pore volume Vpor (solidification of the microstructure due to C-S-H volume increase). The effect of the water shortage in the pore system on the kinetics Q2 can be calculated from the actual free pore wall area Awat, relative to the total pore wall area Apor, in the system. This ratio can also be calculated from the minimum pore diameter and maximum pore diameter ^por in the system, and a pore diameter that represents the largest diameter of those pores that are still completely filled with water ^wat according to

Q =

wat («) _ Фwat,a - Ф0 Фрапа Apor(a^ фрог,а -фо ф

2 A (а) Ф -Ф. ' Ф (15)

The pore diameters in equation (15) change with progress of the hydration process and, implicitly accounts for the influence of the pozzollanic reaction and related morphological changes, which make Q2 also a function of the degree of hydration aPj of the blended system. A pore size distribution model (Fig. 4) is used to calculate these diameters during hydration and porosity measurements can be employed to validate this model [5, 23, 40]. The last reduction factor Q3 refers to the amount of water still available in the pore system to accommodate the ions that dissolve in the capillary pore water during the chemical reaction of cement and pozzolanic phases, and accounts for the balance between the possible supply and precipitation of ions of the reaction products. The reduction of the amount of water in the hydrating paste is considered to cause a shortage of water for the ions to accommodate in the microstructure and react to other phases (for more detailed information reference is made to [5]). This so-called "water shortage concept" affects the kinetics and causes a reduction of the rate of hydration due to the consumption of water by cement and pozzolanic hydration and can be calculated according to [5, 23, 40].

wbr - w -ac ■ m - w -а, ■ m __c j c_p j_p

3 wbr

(16)

where wbr is the water to binder ratio, mc and mp the mass fractions of cement and pozzolans, respectively. The factors wc and wp refer to the that particular water to cement ratio at which either the cement or the pozzolanic phase is completely hydrated (a=1). For both cement and pozzolan a factor 0.4 has been adopted [5, 23 40]. With the basic rate equation as described in a simplistic form in eq. (13), and which accounts for the water dependency (Q1 to Q3), the temperature dependency of the reactions F1 (Arrhenius equation), and the dependency of the state of reaction, i.e. phase boundary or diffusion controlled, an integrated particle kinetics approach has been achieved that can be applied to all particles in the system. The actual position of particles represents the initial state of the particle structure (Fig. 1) and includes both cement and pozzolanic particle fractions. The numerical algorithm should account for the incremental increase of the hydration process, associated changes in microstructure, state of capillary water and chemistry.

Based on the theory presented so far, the formation of a 3D virtual microstructure can be simulated for Portland cement systems (Fig. 3 a) or blended pozzolanic systems (Fig. 3b). Figure 3 a shows a 3D virtual microstructure with a water to cement ratio (wcr) of 0.45 and a Portland cement with specific surface area of 400 m /kg, after 28 days of hydration. In this microstructure yellow represents the outer C-S-H gel and red the inner C-S-H gel, while the remaining original cement grain is colored grey. Apart from these solid phases a capillary pore structure can be identified from the virtual microstructure and is colored blue. This pore structure forms the main "infrastructure" of a (dissolved) matter that may enter or move through the internal microstructure. A water filled capillary pore system, in general, may accommodate chlorides to move or to bind inside a microstructure. However, in case of CO2 sequestration at extreme conditions, i.e. high temperature and high pressure, both water filled capillary pores and empty pores are of major importance. In water filled pores, CO2 may react with calcium hydroxide in the pore water introducing a carbonation

reaction that leads to a de-passivation of the pore water and to a potential leaching of the microstructure. Moreover, empty pores are a way for the CO2 to penetrate the microstructure and possibly escape or introduce harmful instabilities to the cementitious sealing material from the interior [8]. In this case, pore continuity, representing the ability of capillary pores to be connected throughout a microstructure [4, 16-18, 30, 34, 37, 40], is a way to assess the quality of the cementitious sealing material for the casing in a wellbore. The numerical model Hymostruc has the ability to calculate the pore continuity of the capillary pores in a virtual microstructure for a given slurry composition. In the next section the background of this method will be explained in more detail.

Pore continuity

Assessing the pore continuity can be done by means of evaluating the digitized 3D array of the capillary pore structure that is embedded inside the virtual microstructure (Fig. 3). With the 3D digitalization of the pore structure, voxels (3D pixels) are introduced that comply with a certain resolution, which is user defined. Fig. 5 shows the sensitivity of the microstructure toward the resolution of the voxels. Fig. 5a shows the 2D front view of the 3D microstructure (Fig. 3 a) at a resolution of 1 pixel per pm and Fig. 5b shows the same 2D front view at a resolution of 3 pixels per pm. When coloring all solid hydration products white, and the remaining capillary pores black, the morphology of the pore structure becomes clearly visible. This image can be used as the basis for the 3D continuity assessment of the capillary pore structure. The digitized 3D structure of the capillary pore network can be considered as a special binary structure that can be evaluated with respect to state each black voxel is connected. For each pore voxel in the system it should be determined how it is connected to its neighboring voxels, i.e. whether it borders to another "pore voxel" or to a "solid phase voxel". This approach enables the possibility to calculate the change of the pore continuity in a 3D microstructure, as a function of the degree of hydration. With solidification of the microstructure due to hydration, the relative volume of the connected capillary pores will decrease and pores might become disconnected. In order to handle this process consistently, a Flood fill algorithm has been implemented into the Hymostruc code. Flood fill, also called seed fill, is an algorithm that determines the areas connected to a given node in a multi-dimensional array [13]. The algorithm takes three parameters: a start node, a target color, and a replacement color. The method evaluates all nodes in the array which are connected to the start node by a path of the target color. With this algorithm, neighbors of a 3D voxel can be found in a 6 direction or in 26 directions configuration. The difference is whether the voxels situated at the corners of a central voxels are also considered as neighbors. For this 2D plane, a 4 or 8 neighbor configuration can be selected for a pixel which differ from the fact that the neighbor pixels at the corners are or are not considered as connected, respectively. Both approaches may give different results for the pore continuity, but this issue should also be considered with respect to the significant differences in computation time. The sensitivity of the calculated pore continuity towards the voxel resolution turned out to be relevant as well. In Fig. 6 a 2D view of a 3D microstructure (100 x 100 pm2) is shown for three different resolutions, i.e. 1, 2 and 3 pixels per micrometer. The images show an increasing sharpness of the microstructural morphology with increasing resolution. The question is, however, how this resolution affects calculation results for the volume of capillary pores that are still continuous over the full depth of a 3D microstructure. This has been examined by calculating the pore continuity for three different microstructures by applying three different resolutions. It should be beard in mind that the resolution determines the smallest pores that can be detected by the algorithm, which means that for a resolution of 1 pixel/pm only pores can be detected from 1 pm and up. Fig. 7, shows the results as a function of the actual capillary pore volume. Results show that the relative pore continuity, representing the actual volume of capillary pores that are still continuous over the initial pore volume, is decreasing with decreasing capillary pore volume. The resolution only shows little influence on the results and show equal tendency. However, in order to demonstrate the dependency of the pore continuity on the water to cement ratio, results for a Wcr of 0.35 and 0.55 are also included in the figure as well. It can

be observed that the pore continuity and the capillary pore volume have the same tendency, however, they follow a different hydration rate. In Fig. 8, an example of the a 3D capillary pore structure is provided for a water cement ratio of 0.45 and a Blaine value of 400 m2/kg, after 1000 hours, 1 year and 20 years. The particle diameters range between 5 and 20 |m and the microstructure rib size is 100 |m. Due to the relatively course cement grain structure, the capillary pore structure is occupying a large portion of the paste and the pores show a connected configuration. The evolution of the degree of hydration for the four Bogue phases is shown in Fig. 9 (kinetic rate constants C3S/C2S/C3A/C4AF: 0.03/0.002/0.136/0.01 |m/h). When applying the pore continuity algorithm to pore structures as provided in Fig. 8, the change of pore continuity can be presented as a function of the degree of hydration or time. Fig. 10 shows the results together with the change of pore phase and solid phase of the developing 3D pore structure as a function of time. The figure shows a reduction of the continuous capillary pores, a decreasing capillary volume and an increasing volume of solid phase, which represents the total volume of hydration products such as C-S-H gel and Ca(OH)2. Fig. 10 shows the relative pore continuity, the pore phase and solid phase as a function of the actual volume of capillary pore water. The figure shows a solidification of the microstructure as a function of the water uptake due to hydration. When combining both Fig. 9 and Fig. 10 it can be seen that the capillary pores do not completely disconnect. About 15 % of the pores is still connected (Fig. 7), and is, therefore, able to act as a medium to transport phases, such as CO2. In order to show the amount of empty pore space that is connected, the capillary pore volume is multiplied by (1-s), where s is the actual degree of saturation. Fig. 11 shows an increasing volume of empty pore space with decreasing capillary pore volume. It turns out that in this case the empty pore space is about half the total pore phase. This means that after hydration the empty pores may have a significant share in the total pore volume of the microstructure.

a) 1 pixel per

um

• « m

b) 3 pixels per |jm

Fig. 5. Digitized 2D front view of the 3D microstructure (see Fig. 3,a) with a) resolution 1 pixel per ^m, and, b) 3 pixels per ^m

1 pixel per ^m 2 pixel per ^m 3 pixels per ^m

Fig. 6. 2D view of microstructure using two different resolutions

& 0.8 -

3

с ■.p

с о

и ал о

QL

Щ

>

го

ш ос

0.6

0.4 -

0.2 -

/// /// / // // / # / ж 1 ppm, wcr 0.45

2 ppm, wcr 0.45

3 ppm, wcr 0.45

✓ y — — — 1 ppm, wcr 0.35

— — 1 ppm, wcr 0.55

0.2 0.4

Capillary pore volume

0.6

0.8

Fig. 7. Relative pore continuity calculated from 3D microstructure for 1, 2 and 3 pixels per ^m (ppm) and for three

different water cement ratios (wcr) after 28 days of hardening

After 1000 hours (as = 0.69)

After 1 year (as = 0.83)

After 20 year (as = 0.94)

1

0

0

Fig. 8. 3D pore structure of virtual microstructure (bulk paste)

C3S

C2S /-------i//^-

C3A

C4AF

JH—

0

0.0001

0.01

1

Time [days]

100

10000

Fig. 9. Evolution of the degree of hydration for the four Bogue phases C3S, C2S, C3A and C4AF

0,0001 0,01 1

Time [days]

' Solid phase

100

10000

Fig. 10. Evolution of relative pore continuity, pore volume and solid phase as a function of time

1

Pore continuity in slurry microstructures

Microstructures of cement slurry pastes that seals off casings in wellbores and that run to deep underground storage reservoirs for CO2 capturing can also be simulated for their extreme downhole temperature conditions. Table 1 shows the chemical compositions of the cementitious material that was taken as input for the simulations and Table 2 shows the slurry design. A 3D microstructure was generated representing the initial basis for the hydration calculation and for the associated effect on the pore continuity. During hydration the capillary pore volume reduces due to the binding with clinker minerals and forms the inner and outer C-S-H gel. The rate of this process is depending on the ambient temperature conditions and comply with the Arrhenius law [5]. The morphology of the outer C-S-H gel turned out to be affected by the increase of temperature, resulting in a denser packing of the outer C-S-H gel around a particle and a lesser growth in outward direction. Because of this, the capillary pore volume may increase up to 10 to 15% [5]. Fig. 12 shows the temperature effect on the relative pore continuity versus the actual capillary pore volume in the system. The ambient temperatures are taken equal to the mix temperature of T= 20 °C, and then follow adiabatic conditions until T = 40 °C or T = 60 °C has reached. From there, the temperature is kept constant. These conditions mimic the mixing of slurry at surface temperature of T = 20°C, and while being pumped down hole, temperature rises up to the imposed temperature level of either T = 40 °C or T=60 °C. In order to show the effect of temperature on the pore continuity more explicitly, the results of a 0.3 water to cement ratio slurry are added to the figure as well. Results show that with higher ambient temperatures pore volumes increase and discontinuity of the capillary pores occur at later age. The temperature sensitivity turns out to be higher for lower water cement ratio pastes. It is worth notifying that hydration may not always lead to a complete disconnection of the capillary pores (Fig. 11 and 12). The continuous pore volume after hydration has ceased is about 20% for the 0.44 Wcr slurry, while for the 0.3 wcr slurry a complete discontinuation of the capillary pore volume occurs. However, the temperature effect seems to coarsen the pore structure, leading the discontinuation to occur at smaller capillary pore volumes. In order to analyze the pore structure of the continuous pore volume in more detail, a pore formula [23] can be applied that relates the maximum pore diameter to the continuous capillary pore volume Vp0re,c, (eq. 17).

^ ^oexpC^) (17)

a

This formula is used for the results of the 0.44 wcr slurry, and taking a pore structure constant a=0.1 and a minimum pore diameter of фо=2 nm 523. The calculated results show the maximum pore diameter of the continuous pores to range from 0.93 pm at the beginning of hydration until 6.5 nm, 7.8 nm to 9.0 nm at a degree of hydration of ac = 0.8, for ambient temperatures of T = 20 °C T = 40 °C and T = 60 °C, respectively. Comparison with experimental data from Justnes et al [22] shows very good agreement. The results can also be used to calculate the permeability kw of the system according to [23]

h _Oo2 + Qg

kw - 64m (18)

where v is the kinematic viscosity of water. Fig. 13 shows the calculated permeability for the three ambient temperatures conditions. The results show a same tendency as the results obtained from literature [17, 22]. The calculated permeability represents a kind of "effective" permeability that is based on only the continuous pore volume in the slurry system.

Table 1

Mass fractions of cement phases from literature

Cement phase Mass fraction 1 [6] Mass fraction 2 [7]

C3S 61-62 63.94

C2S 15-16 15.84

C3A 0.5-1.5 0.57

C4AF 15-16.5 11.33

Table 2

Slurry composition used in the simulation

Cement phase Quantity

Water cement ratio 0.44

Blaine 500 m2/kg

Temperature 20 °C - 40 °C - 60 °C

Pressure Atmospheric

C3S/C2S/C3A/C4AF 65.5/15/1.5/11

Durability of slurry microstructure

Capturing CO2 gasses in exhausted oil or gas reservoirs requires a very robust solution for the cement slurry that seals off the space between the outer casing and inner wellbore. Oil well cement slurries are especially designed to fulfill this purpose. However, bearing the extreme temperature and pressure conditions that prevails in the vicinity of deep underground reservoirs is a challenge for advanced cement slurries designs. In case of CO2 capturing, most severe degradation mechanism that may harm the microstructure integrity is the carbon dioxide itself by potentially introducing carbonation [32]. Carbonation is a durability reducing phenomenon that is commonly observed in cementitious systems exposed to CO2 [23]. Most important deterioration mechanism is the potential damage to the internal microstructure due to the dissolution of calcium hydroxides via the pore water, which causes the CO2 to react with calcium and hydroxyl ions to form calcium carbonate. The overall reaction can be described as follows:

Ca(OH)2(s) + CO2(g)^CaCO3(s) + H2O. (19)

Although the stable calcium carbonate may initially result in a lower permeability, dissolution of precipitated calcium carbonate may lead to an increased capillary porosity and a disintegration of the

microstructure [42]. Precipitation of calcium carbonate, which is less soluble than portlandite can provide a temporary, less permeable front to the CO2 attack. However, after depletion of portlandite, pH drops and calcium carbonate dissolves more due to the effect of pH on speciation of carbonate ions. When the pH drops below 11, the CO^ concentration decreases, and bicarbonate, HCO3, begins to dominate. The hydrolysis of

carbonate decreases the concentration of CO^, which pulls the solubility equilibrium (CaCO3^ CO^ +

2+

Ca ) to the right making CaCO3 more soluble.

CaCO3(s) + CO2(aq) + H2O ~ Ca(HCO3)2(aq). (20)

2 2+

The final effect of the hydrolysis of CO33 on Ca solubility as a function of CO2 partial pressure (at

25 oC) can be demonstrated by following calculated points (Ksp = 4.47*10-9 has been taken for the calculation): 6.62 10-4 mol/L, 6.58 10-3 mol/L and 1.42 10-2 mol/L for 10-3 atm (pH=8), 1 atm (pH=6) and 10 atm (pH=5.3), respectively. This phenomenon is considered to be a mechanism that may be most harmful to the structural integrity of the sealing performance of cement slurries used in CO2 sequestration. After depletion of CaCO3 (that buffered the pH) the remaining unhydrated cement particles and the CSH gel are decalcified (dissolved). Experiments conducted with a continuous flow of acidified carbonated brine observed extensive degradation of the samples, in contrast to the experiments with static external conditions [26]. Field experience in oil recovery fields that were enhanced by CO2 injection has shown that wells can maintain a seal for decades often with a limited depth of CO2 attack [27]. This indicated that the degradation processes involved proceed slowly. Long-term experiments are still underway to study the kinetics of cement degradation under these conditions. The existence of a solid phase assemblage with clearly defined dissolution fronts [26] may be explained by instantaneous dissolution [36], i.e. establishment of the local liquid equilibrium concentrations.

Thus, in case of static external conditions, the kinetics of the dissolution reaction is governed by a diffusion process, because the diffusion rates are much slower than those of the chemical reactions. Detailed simulations can be conducted that show the sensitiveness of a microstructure towards the evolution of hydration products in the long run. For this, first, chemical reactions have to be calculated for each individual cement grain that contains the four main clinker minerals as represented by the Bogue phases. The hydration reactions comprise the silicate reactions for the silicate minerals C3S and C2S and the aluminate-ferrite reactions which concern the C3A and C4AF minerals. From these components the reaction products are calculated generating the C-S-H gel and the portlandite Ca(OH)2 formations.

The production of portlandite can be indicative for a potential dissolution from the microstructure due to carbonation, which may cause an increase of the porosity and encourage disintegration of its internal structure. Fig. 14 shows the growth of the portlandite with elapse of time for the three different temperature conditions considered. A potential dissolution and distraction of portlandite from the microstructure or a potential ingress of CO2 into it, or even through the empty pores of the microstructure, may be simulated with a 3D flow algorithm [15]. A flow model is being implemented in the Hymostruc model that enables the possibility to simulate the transport of harmful matter [15] through the evolving microstructure.

In case of CO2 it should also be noticed that the extreme downhole conditions may significantly affect the rate of the chemical processes. High temperature and high pressure situations will increase the rate of the degradation processes at least according to the Arrhenius law and where the high pressure will even further increase the rate at which degradation may take place, a potential harm to durability of the cementitious sealing material may appear.

Effect of ITz on sealing performance

In the occasion that sand or other non-reactive fillers or aggregates are added to a cement slurry, interfacial transition zones (ITZs) are introduced inside the cementitious microstructure. These ITZs are

typically identified as transition zones between aggregate surfaces and the cement paste and are relatively porous zones with an average thickness that ranges from 10 to 30 |im, depending on the cement fineness and the water cement ratio [15]. These relatively high porosity zones are created because of the non-optimized packing of the cement particles situated in close vicinity of aggregate surfaces, called the "wall-effect". These zones of limited packing of cement particles around aggregates may, once inter-connected, influence the permeability, and with this, the sealing performance of a cementitious slurry system significantly. The ability of these ITZs to act as an internal inter-connected porous network depends on the ratio between the permeability of the bulk paste (for instance Fig. 13) and the potential permeability of the connected ITZ network.

Experimental tests of rapid chloride migration have shown that for Portland Cement (PC) based mortars the ingress front of chlorides was quite homogeneously (Fig. 15a). The bulk past permeability clearly dominates over the connected ITZ network permeability. However, other tests conducted on similar samples using Blast Furnace Slag Cement (BFSC) based mortars showed a completely different pattern (Fig. 15b,c). The denser microstructure of BFSC systems clearly resulted in a dominating ITZ network permeability over the bulk paste permeability. The chlorides turned out to move through the microstructure making use of the inter-connected ITZ networks as an alternative route, and by-pass the denser and less permeable microstructure of the bulk paste.

A closer look to the ITZ really confirms that chlorides may accommodate in the porous ITZs and confirm the ability to act as an internal network of "channels" (Fig. 15). Numerical simulations with the pore continuity model [15] can be done to analyze the pore connectivity of the ITZ relative to the bulk paste. The continuity of a capillary pore system has been analyzed for a ribbon paste microstructure (rib 100 |im), which represents a paste volume in between two aggregate particles. The results show the continuity of the capillary pore volume over three different thicknesses of the ITZ starting from the aggregate surface, i.e. 0-5 |im, 010 |im and 0-20 |im, as well as pore continuity of the bulk paste, i.e. 20-80 |im (Fig. 16 and 17). At hardening large differences occur that go up to a factor 10. These differences indicate that the pore continuity of the ITZ regions and bulk paste may behave very different in an internal capillary transport system.

0,8 -

с О

.Ü 0,6 H

2

га 0,4 A

£ 0,2 -

Relative pore continuity

Pore phase

^^ Solid phase

Empty pore

0,2 0,4 0,6

Capillary pore volume [cm3/cm3]

0,8

Fig. 11. Development of relative pore continuity, pore phase and solid phase versus the capillary pore volume

wcr = 0.44

T=20 C T=40 C T=60 C T=20 C T=60 C

0,2 0,4 0,6

Capillary pore volume [cm3/cm3]

Fig. 12. Relative pore continuity versus capillary pore volume for Wcr=0,3 and 0.44, and for T=20, 40 and 60°C

1

0

0

Time [hours]

Fig. 13. Permeability versus hydration time: effect of curing conditions

Time [hours]

Fig. 14. Portlandite versus time

a)

Fig. 15. Rapid chloride migration tests with ingress profiles of Portland cement concrete (a) and Blast furnace slag cement concrete (b and c)

Fig. 16. Close up of an aggregate particle with chlorides observed in the ITZ

1 0.9

0.2 0.1 0

0.4 0.6 0.!

Degree of hydration

2

Fig. 17. Left; State of capillary water over the ribbon paste (wcr 0.4, Blaine 400 kg/m ) after half a year of hydration, Right; Pore connectivity over the ribbon paste (ITZ and bulk paste)

Conclusions

A numerical study has been performed to scrutinize the pore continuity of hardening cementitious slurry systems and to assess durability. Results are calculated for a pore structure embedded in a 3D virtual microstructure which is used to assess the pore continuity of a capillary pore space in an evolving cementitious slurry quantitatively. A 3D virtual microstructure of cement slurry is digitized and the pore structure is analyzed in terms of connectivity and permeability. The model results show good agreement with literature data. A disintegration of the microstructure could be predicted with the model and quantifying durability. Adding aggregate or non-reactive fillers to a cementitious slurry is introducing potential alternative routes for moisture or dissolved matter such as CO2, to move through a cementitious system. A flow model is currently being implemented enabling the possibility to simulated 3D flow of harmful species through an evolving microstructure and accounts for a change of the pore structure, and with this the permeability. Future research efforts will focus on the effects of partly-saturated liquid water conditions and coupled reactive-transport investigated by numerical models implemented directly on the virtual 3D microstructures.

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Note: Most parts of this article have also been published as E. A. B. Koenders, W. Hansen, N. Ukrainzcyk and R. D. Toledo Filho, Modeling Pore Continuity and Durability of Cementitious Sealing Material, J. Energy Resour. Technol. 136(4), 042906 (Oct 13, 2014) (11 pages), http://dx.doi.org/10.1115/L4028692

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Э.А.Б. Коендерс

ЭДУАРДУС АЛОЙСИЗ БЕРНАНДУС КОЕНДЕРС - профессор, факультет гражданского строительства и геодезии (Технический университет Дармштадта, Германия). Офис 3, 209, ул. Франциска Брауна, г. Дармштадт, 64287, Германия. E-mail: [email protected]

Моделирование поровой системы и цементной суспензии при высоких температурах

В данной работе представлены результаты численного моделирования поровой системы, проницаемости и прочности цементных суспензий для случая секвестрации углерода.

Модель гидратации Hymostruc применяется для моделирования и визуализации виртуальных 3D микроструктур, которые используются, чтобы продемонстрировать вклад капиллярных пор капиллярной поровой системы в формирующейся цементной микроструктуре. Как только капиллярные поры забиваются из-за процесса гидратации, движение CO2 через микроструктуру прекращается, и это защищает цементную суспензию от расслоения. Поэтому поры капиллярной поровой системы в процессе гидратации микроструктуры важны для уплотнения цементной суспензии, что является основной целью для проектирования состава суспензии. Моделирование суспензий выполнено при температурах 20 °С, 40 °С и 60 °C для оценки долговечности при движении CO2. Заполнители и связанные с ними межфазные переходные зоны в суспензии приводят к образованию дополнительных капиллярных пор. Анализ поровой системы показывает важность численного моделирования капиллярной поровой структуры внутри формирующейся цементной микроструктуры в части повышения ее непроницаемости и долговечности.

Ключевые слова: бетон, цемент, пора, микроструктура, проницаемость, прочность.