NISTIR 7026

Condition Assessment of Concrete Nuclear Structures

Considered for Entombment

Kenneth A. Snyder
NISTIR 7026

Condition Assessment of Concrete Nuclear Structures

Considered for Entombment

Kenneth A. Snyder

Materials and Construction Research Division

Building and Fire Research Laboratory

Sponsor:

U.S. Nuclear Regulatory Commission

July 2003

UNITED STATES OF AMERICA

DEPARTMENT OF COMMERCE

U.S. DEPARTMENT OF COMMERCE

Donald L. Evans, Secretary

TECHNOLOGY ADMINISTRATION

Phillip J. Bond, U ndersecretary of Commerce for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

Arden L. Bement, Jr., Director
Condition Assessment of Concrete Nuclear

Structures Considered for Entombment

K.A. Snyder

July 14, 2003

VERSION: 20030108:1545

1
Abstract

This Report summarizes work to date on a project for the U.S. Nuclear Regulatory

Commission to develop guidelines for the chemical and material assessment of an existing

structure considered for entombment. In addition, a rational means for performing a

probabilistic calculation are addressed. This report is limited to materials issues as they

relate to the overall performance of the structure. Structural issues such as the presence

of cracks and the location of rebars are discussed only in the context of their effect on

transport.

The assessment is composed of a preliminary material assessment, followed by a probabilistic calculation of the service life. The material assessment will rely heavily on both

the permeability of the intact concrete and the characterization of any existing cracks.

The contribution of cracks to transport is considered independently from the intact concrete, and the bulk properties (intact concrete plus cracks) are estimated by combining

the two effects using a composite model. Another aspect of the condition assessment of

the concrete barrier is the evaluation of nondestructive techniques (NDT) as candidates

for characterizing existing cracks, and other transport pathways, in the entombment

structure. This aspect will be addressed in a companion report.

An important component of the effort to estimate the concrete material properties is

the probability density function used to characterize the distribution in their expected

values. The “tail” of the distribution has the greatest influence on the overall performance, and, therefore, it should receive attention in the analysis. A number of suitable

distributions are considered for their use in characterizing various concrete material properties.

iii
Contents

Abstract iii

1 Overview 1

2 Condition Assessment 3

2.1 Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 Expected distribution . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.2 Direct measurement . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.3 Indirect measurement . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Water:Cement Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3.1 Expected distribution . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.2 Direct measurement . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.3 Indirect estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4.1 Expected distribution . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4.2 Direct measurement . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4.3 Indirect estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.5 Formation Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.5.1 Expected distribution . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5.2 Direct measurement . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5.3 Indirect estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.6 Pore Solution Composition . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.6.1 Direct measurement . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.6.2 Indirect estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Crack Characterization 10

3.1 Composite Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Crack Parameter Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 Extensive Cracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5 Subsequent Crack Initiation . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5.1 Mechanically induced cracking . . . . . . . . . . . . . . . . . . . . 15

3.5.2 Chemically induced cracking . . . . . . . . . . . . . . . . . . . . . 15

3.6 Other Major Transport Pathways . . . . . . . . . . . . . . . . . . . . . . 15

4 Sampling Plan 16

4.1 Stratified Random Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2 Probabilistic Calculation Requirements . . . . . . . . . . . . . . . . . . . 16

4.3 Sample Size Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.3.1 Chi-square distribution . . . . . . . . . . . . . . . . . . . . . . . . 18

4.3.2 Student t-distribution . . . . . . . . . . . . . . . . . . . . . . . . 19

iv
4.3.3 Adjusted Distribution . . . . . . . . . . . . . . . . . . . . . . . . 20

4.4 Analytical Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.5 95-th Quantile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Example 25

5.1 Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2 Concrete Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.3 Material Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.3.1 Capillary Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.3.2 Formation Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.3.3 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.4 Pore Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.5 Portlandite Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.6 Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.7 Negative Internal Hydrostatic Pressure . . . . . . . . . . . . . . . . . . . 30

5.8 Exposure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.9 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.9.1 Seawater Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.9.2 Brackish Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Summary 35

References 36

v
List of Figures

1 Crack model schematic for a concrete slab with span L, depth h, and

containing m cracks, each having width w and penetrating a fraction α

into the element. ko is the permeability of the uncracked concrete, and kc

is the permeability of each crack. . . . . . . . . . . . . . . . . . . . . . . 10

2 The ratio of the bulk permeability kb to the intact concrete permeability ko

plotted as a function of crack penetration ratio α. The crack parameters

are found in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Schematic of a stratified random sampling plan. A structural element

is subdivided (dashed lines) into subsections. Samples (filled circles) are

taken from random locations within each subsection. . . . . . . . . . . . 17

4 Chi-square distribution gk(v) for different degrees of freedom k. . . . . . 19

5 Student t-distribution hk(t) for 1, 2, and 5 degrees of freedom k. The

dashed line is the N(0, 1) distribution. . . . . . . . . . . . . . . . . . . . 20

6 Modified material parameter distribution to account for finite sample size.

The degrees of freedom k are labeled along the ordinate. The dashed line

denotes the N(0,1) distribution. . . . . . . . . . . . . . . . . . . . . . . . 22

7 y.95 as a function of the degrees of freedom k. . . . . . . . . . . . . . . . 23

8 Example scenario showing 100 cm thick wall with 10 cm diameter reinforcement gallery in the center. . . . . . . . . . . . . . . . . . . . . . . . 25

9 Response of concrete exposed to seawater: (a) Probability of failure Pf

as a function of the failure time Tf ; and (b) The logarithm of hydraulic

conductivity K as a function of of time t. The error bars represent one

standard deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10 Response of concrete exposed to brackish water (salinity: 3.5 g/kg): (a)

Probability of failure Pf as a function of failure time Tf . Permeating

systems shown with solid lines and the addition of cracks denoted by the

dashed line in proximity to the solid line system; and (b) The logarithm of

hydraulic conductivity K as a function of of time t (mean values shown);

and (c,d) The logarithm of the hydraulic conductivity K, as in (b). Error

bars represent one standard deviation. . . . . . . . . . . . . . . . . . . . 33

vi
List of Tables

1 The 95-th quantile y.95 for the distribution p(y; x, s). Also shown are the

Student-t statistic t0.95 and the ratio of the Chi-square statistic to the

degrees of freedom χ

2

.95/k. . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Input parameters to the ACI 211 mixture design procedure. . . . . . . . 26

3 Concrete mixture design based on ACI 211 for normal strength concrete.

The corresponding w/c is 0.45, and the estimated slump is 51 mm. . . . 26

4 Seawater composition based on Schlesinger (Table 9.1, p. 263). The brackish water has a salinity of approximately 3.5 g/kg. . . . . . . . . . . . . . 31

vii
1 Overview

Entombment may be considered as a decommissioning option for reactor components

and other nuclear structures so that the licenses for facilities on which these structures

reside may be terminated. Prior to the entombment process, the nuclear facility permanently ceases operations and spent fuel is removed. After preliminary decommissioning

activities are completed, activities that may include the removal of a small volume of

highly radioactive material, remaining radioactively contaminated components may be

entombed within the structure using an engineered barrier system that could include

filling the structure with cement, absorbant grouts, or infills. Entombment may reduce

exposure to the workers and to the public during decommissioning because only a small

volume fraction of the structure has to be transported to a waste isolation facility. An important consideration for decommissioning is that the existing concrete nuclear structure

is a primary barrier between its remaining contents and the environment.

As a consequence, however, the entombed nuclear concrete structure may be the final

barrier between the contaminated material and the environment. Therefore, a careful assessment of the future performance of the entombed concrete structure is vital to assured

isolation. The assessment involves characterizing the existing condition of the concrete

structure and the chemistry of the ground water within the surrounding soil. The condition of the existing structure will be characterized primarily by the relevant transport

coefficients, and the geometry of the critical concrete elements. The surrounding ground

water will have to be tested to determine the concentration of each dissolved ionic species.

The final service life estimate will be probabilistic, based on parameter uncertainty.

Therefore, it will be important to fully characterize the uncertainty distribution of the

relevant transport coefficients. This will require sufficient sampling to assure that the

“tails” of the distribution are known to a satisfactory level, for the “tail” of the probable

outcomes will depend directly on them.

As will be shown, it will be vital to carefully examine the structure for the existence

of cracks. When advective (Darcy) flow needs to be considered, cracks are an important

factor in estimating the total transport through the concrete. Three crack parameters

are significant: crack width at the surface, the number of cracks per unit length, and the

depth of crack penetration. Although the permeability of a crack is a quadratic function

of the crack width, the depth of crack penetration is, by far, the most important crack

parameter determining the overall permeability of the structure.

In addition to the existing condition, one should also consider the possible changes

in the structure and its environment. Changes in the loading within the structure may

propagate existing cracks, or induce the formation of new cracks. Changes in the environment may induce degradation mechanisms that can create internal stresses. These

possibilities, and their effect on transport through the structure, must be considered.

Subsequent to the assessment of the concrete and the characterization of existing

cracks, one can make an estimate of the service life based on the 4SIGHT computer

program [1, 2]. The program attempts to consider as many material and environmental

properties as possible to estimate the time until the onset of a degration mechanism.

1
The service lifetime estimation is based on a reactive transport algorithm that correctly

models the interaction of ions within the pore solution [2,3]. Because the expected service

lifetime of massive structures is expected to be measured in decades or centuries, the time

between the onset of degration and the development of sufficient stress to produce failure

is assumed to be an insignificant proportion of the overall service lifetime.

2
2 Condition Assessment

The objective of the concrete structure condition assessment is to quantify the relevant

material properties required to estimate the remaining service life. The primary cause

of premature failure of a concrete structure is chemically induced mechanical degradation. Generally, the deleterious chemical reactions are due to diffusion of ionic species.

Therefore, assessment of the concrete depends on its present condition and the spatial

distribution of diffusing ionic species.

In addition to characterizing the present condition of the structure, the assessment

must also estimate the uncertainty in the relevant material parameters. This involves

choosing a probability distribution that accurately characterizes the dispersion in values

of the parameter in question. Given that the chosen underlying probability distribution

is suitable for the particular property, the parametric approach has the advantage of

reducing the number of samples required for characterizing the range of possible values

expected; in effect, it becomes easier to characterize the “tails” of the distribution.

In the following, each of the relevant material parameters is described and methods are

reviewed for measuring or estimating the values and uncertainties, and for characterizing

the underlying distribution. Arguments are given for choosing particular distributions to

characterize a parameter.

2.1 Probability Distribution

Probabilistic calculations require random values for the material parameters. The user

must choose a probability density function (PDF) for characterizing the dispersion of

expected values. Some distributions have longer “tails” than others, so the choice of the

PDF is important because the failure scenarios likely depend upon the extremes in the

values of one or more of these parameters.

Currently, there are four PDFs to choose from: Dirac delta function, uniform (top hat)

distribution, normal (Gaussian) distribution, and lognormal. The Dirac delta function

allows the user to specify a fixed value for a parameter [4]. The uniform distribution is

for properties that can be bounded. The normal distribution is for properties that vary

over a factor of two or less. The lognormal distribution is for properties that vary over

orders of magnitude [5].

2.2 Geometry

Although material parameters are critical to the assessment, there are two structural

geometrical parameters of interest: concrete element thickness and the location of steel

components such as reinforcement. The concrete element thickness determines the spatial boundary conditions. There are methods for determining this dimension: direct

observation, nondestructive techniques (e.g., sound wave propagation), and construction

schematics. The location of steel reinforcement, if present, effects the time until the

possibility of corrosion initiation. In the presence of chloride ions, steel corrosion is likely

when the chloride concentration in the pore solution exceeds a critical concentration.

3
2.2.1 Expected distribution

The geometrical quantities of interest are the concrete element dimensions and the location of steel elements that, with sufficient corrosion, could lead to mechanical failure

of the structure. Because concrete elements and the location of steel components were

established from technical construction plans, the actual dimensions should be relatively

close to the designed values. Therefore, one would expect that the underlying distribution

should be normal, with relatively little variance.

2.2.2 Direct measurement

The most reliable determination of concrete element dimension is direct physical observation. These observations can be compared to the original plans for compliance

determination.

2.2.3 Indirect measurement

For cases involving inaccessible concrete elements, nondestructive techniques may be suitable alternatives [6]. As for direct measurement, the nondestructive testing results can be

compared to the original construction plans for independent verification. Alternatively,

the construction plans, in the absence of any measurements from the structure, could

be used to establish dimensions. This approach, however, gives no assurances that the

dimensional requirements have been met.

2.3 Water:Cement Ratio

At the time the concrete mixture is formulated, the most important parameter controlling

the overall properties of the concrete is the water:cement (or water:cementitious for the

case of concretes using supplemental pozzolanic materials) ratio. As such, knowledge of

this ratio can be very helpful for inferring material properties that might not otherwise be

obtainable. Water:cement ratio controls the concrete compressive strength, permeability,

and diffusivity.

Concrete for large projects such as a nuclear facility typically is batched at a plant

using computer controlled delivery systems. Although attempts are made to produce a

consistent product, variations in the water:cement ratio are to be expected. One could

expect that a concrete producer with stringent quality controls should be able to produce

a concrete with minimal variation in the water:cement ratio.

Variations in the water:cement ratio can arise during both the batching and the construction. There naturally will be small variations in the batching volumes, resulting in

variations in both the aggregate volume and water:cement ratio. During normal concrete

production, however, the concrete trucks are rinsed clean with water before the next load.

While on transit to the construction site, and while waiting at the construction site, the

mixture may lose water due to evaporation. Further, if the workability of the concrete

has decreased, the truck driver may feel compelled to add water to the mixture to regain

the lost workability.

4
2.3.1 Expected distribution

A number of possible scenarios exist that may lead to variations in water:cement ratio.

According to the American Concrete Institute (ACI), the tolerance for total water content

is 3 % and the tolerance for cementitious materials ranges from 1 % to 4 % depending

on the scale limits [“Standard Specifications for Tolerances (ACI 117-90),” in Part 2 of

Ref. [7]]. Therefore, the water:cement ratio variability should be in the range 3 % to 5 %.

At these percentages, it would seem plausible to characterize the water:cement ratio by

a normal distribution.

The shortest service lifetimes will likely coincide with the largest possible values for

water:cement ratio. These larger values lead to greater porosities, larger permeabilities

and lower compressive strengths, and more rapid transport of aggressive ionic species.

2.3.2 Direct measurement

Direct measurement of water:cement ratio of a fully hydrated concrete specimen is a

difficult process. It typically requires a scanning electron microscope (SEM) analysis of

the hydrated paste within the concrete. In addition, certain assumptions must be made.

Fortunately, an easier way to characterize the variability in the water:cement ratio, and

also the overall quality, is to make compressive strength measurements. Compressive

strength measurements can be obtained from a straightforward test procedure and are

the default means of quality control at the time of construction. Moreover, for a nuclear

structure, the compressive strength measurements at the time of construction should have

been recorded, and can serve as an indirect measurement to complement compressive

strength testing at the time of entombment.

2.3.3 Indirect estimation

It is difficult to estimate the water:cement ratio without either the original specified mixture design or strength tests. The water:cement ratio could be inferred from permeability

measurements, but the relationship is not as reliable as for compressive strength.

2.4 Permeability

The permeability of the intact portion of the concrete is probably the single most important parameter for assessing the future performance of an engineered barrier system.

Typically, the permeability of concrete is on the order of 10−17 m2to 10−18 m2. Although

the measurement of concrete permeability has been reported in the concrete materials literature, measurements on specimens with low water:cement ratios having exceedingly low

permeabilities might have to rely on novel dynamic techniques developed for geological

research [8, 9].

The permeability of concrete is a strong function of the water:cement ratio (w/c).

Unfortunately, even at a fixed value of w/c, the measured permeability of various concretes can still vary over orders of magnitude [10]. Therefore, although knowing the w/c

5
of the concrete used in the structure is helpful, predictions based on published data still

contain uncertainty that spans orders of magnitude.

The sample-to-sample variation within a given structure is still unpredictable and

must be measured directly. Here, the analyst must be careful when choosing the proper

PDF. A small number of measurements may not exhibit sufficient skewness that would

help to distinguish between a normal and lognormal distribution.

2.4.1 Expected distribution

Permeabilities can span orders of magnitude, even for repeated measurements on samples

with nominally the same water:cement ratio. Therefore, the lognormal distribution is the

preferred candidate for characterizing the expected range of values used in a probabilistic

calculation. The parameters of the distribution can be determined by taking the (base10) logarithm of the measured permeabilities and then calculating the mean and standard

deviation of the logarithms.

2.4.2 Direct measurement

Direct measurements of permeability are divided between equilibrium and dynamic measurements. Equilibrium measurements typically consist of applying a hydrostatic pressure difference across a specimen and measuring the flow rate through the sample. The

analysis and estimate of the permeability is very straightforward from the experimental values. Unfortunately, for very low permeabilities (< 10−18 m2

), this approach has

practical limitations.

The dynamic approach uses a slightly different experimental setup [9], and requires

more sophisticated analysis [8]. Reservoirs are located at both ends of the sample and

the pressure in one reservoir is increased by a step function. As the liquid flows through

the specimen, the pressure in both reservoirs approaches an equilibrium value. The

permeability of the sample is estimated from the rate of pressure change in both reservoirs.

This method has been used on geological samples with permeabilities below 10−20 m2.

2.4.3 Indirect estimation

Indirect estimates of the permeability are not recommended. There is no single independent material parameter from which one can make a reliable estimate of the permeability.

Only by direct observation can one make assurances that the distribution of possible permeabilities is characterized sufficiently well.

2.5 Formation Factor

The formation factor is a useful material property for characterizing both diffusive and

advective transport in cementitious systems. The formation factor F is the ratio of the

pore solution conductivity σp to the concrete conductivity σc:

F =

σp

σc

(1)

6
This approach has the advantage that conductivity is a relatively easy and accurate

experimental measurement.

2.5.1 Expected distribution

Even though typical values for the formation factor of concretes range from 100 to 1000,

the values measured from a single structure may vary over a relatively small range. Only

a careful analysis of the data will reveal whether the appropriate distribution should be

the normal or lognormal distribution.

2.5.2 Direct measurement

Direct determinations of the formation factor consist of separate tests for the pore solution

and the concrete conductivities. The pore solution conductivity can be determined from

samples obtained from pore fluid expression [11]. The pore solution conductivity can

then be measured directly using a solution conductivity meter. Otherwise, elemental

techniques such as ion chromatography can be used to determine the concentration of

majority ionic species such as OH−, K+, and Na+, and the conductivity estimated from

these concentrations.

The concrete conductivity can be most easily measured using the apparatus for the

ASTM C 1202 Rapid Chloride Test. It has been shown that the instantaneous current is

proportional to the sample conductivity, to within an acceptable uncertainty [12]. After

the concrete conductivity measurement, pore expression can be used to extract a sample

of the pore solution, and the pore solution conductivity can be determined as described

above.

2.5.3 Indirect estimation

There are two indirect methods for estimating the formation factor. The first uses a

model such as the NIST microstructural model [13] to estimate the formation factor from

the water:cement ratio and degree of hydration. The other makes a direct measurement

of the bulk conductivity (because it is a simple test) and estimates the pore solution

composition. The indirect method that relies on a model for the formation factor is less

reliable because the results can be a function of the types of supplementary cementitious

materials (fly ash, silica fume, etc.) present.

The second indirect method contains less uncertainty because the variability in the

pore solution conductivity is fairly small. For ordinary portland cement pastes, the pore

solution conductivity typically varies between 10 S/m and 20 S/m. The addition of

supplemental cementitious materials can have an effect, but the conductivity will still

typically be above 5 S/m.

2.6 Pore Solution Composition

The rate of ionic transport in porous media can depend upon the chemical composition

of the pore solution. Therefore, a meaningful assessment requires characterizing the pore

7
solution composition. Fortunately, the pore solution is typically composed of only a few

inorganic ionic species: OH−, K+, Na+, Cl−, SO2−

4

, Mg2+, etc. These species can be

detected using a number of established analytical techniques such as atomic absorption

and ion chromatography.

2.6.1 Direct measurement

The direct assessment of the pore solution composition is a simple analytical chemistry

problem, but a more difficult physical testing problem. The most direct means of extracting a sample of the pore solution is by pore expression. Unfortunately, an aged specimen

exposed to atmospheric conditions presents experimental difficulties: it has relatively

little pore solution remaining, and atmospheric exposure can evaporate a portion of any

remaining pore solution. If, however, a sample can be obtained from within the concrete

element, the sample will probably be saturated to its maximum extent.

2.6.2 Indirect estimation

The best means of estimating the pore solution is to use a model such as that by Taylor

[14]. The estimate based on the Taylor model can be simplified by assuming that at

30 years the individual phases have completely hydrated, as is allowed by the initial

water:cement ratio.

The estimate largely depends on the oxide content of the cement and the initial

water:cement ratio. From the oxide content, one can estimate the total number of moles

μT of a particular element per 100 g cement. From this estimate, Taylor assumes that

the number of immediately soluble moles μsol is 35 % of the total sodium content and is

70 % of the total potassium content.

For water:cement ratios greater than approximately 0.40, the cement paste can completely hydrate because there is sufficient water available initially and there is sufficient

initial capillary pore volume to accommodate the net increase in volume of solids produced during hydration. For a 30 year-old structure with a water:cement ratio greater

than 0.40, the Taylor model assumes that all of the available sodium and potassium will

have been released. The maximum degree of hydration αmax for an ordinary portland

cement (OPC) paste is a function of water:cement ratio :

αmax = MIN(2.5, 1)

The number of moles released μr for an element is then a function of the total number

of moles μT and the immediately soluble moles μsol:

μr = μsol + αmax (μT − μsol)

This relationship does not account for the different rates of hydration for the cement

phases.

The mass of water (kg) per 100 g cement is the water:cement ratio divided by 10.

The molality mi of the i-th ionic species is the ratio of moles of element released μr to

8
the mass of water, per 100 g cement:

mi =

μr

/10

The sulfate content is approximated from the molalities mNa and mK of sodium and

potassium, respectively, using an empirical relationship [14]:

mSO4 ≈ (0.06 kg/mol) (mNa + mK)

2

(2)

These three species, along with a sufficient hydroxide content to ensure electroneutrality,

account for the great majority of the ionic species in cement paste pore solution.

9
αh

(1− )h α

L−mw mw

o

o

c

k

k k

Figure 1: Crack model schematic for a concrete slab with span L, depth h, and containing

m cracks, each having width w and penetrating a fraction α into the element. ko is the

permeability of the uncracked concrete, and kc is the permeability of each crack.

3 Crack Characterization

The properties of intact, uncracked concrete are typically sufficient to ensure isolation of

radionuclides for hundreds of years. The presence of cracks, however, can dramatically

increase the rate of radionuclide transport. The permeability of uncracked concrete is

typically 10−18 m2. A crack composed of plane, parallel walls separated by 10 μm (a

very small crack) has a permeability of approximately 10−11 m2, a factor of 107larger

than the value for the uncracked concrete. Even after considering the area fraction of

the concrete occupied by the crack, the total advective transport can still be greater by

a factor of 10. Therefore, it is imperative to carefully characterize the geometry of any

cracks present in the structure.

An investigation of the effect cracks have on the total transport was undertaken

to evaluate which, if any, of the crack parameters has the greatest influence. Three

crack parameters were considered: crack width, number of cracks per unit length of

span, and the depth of penetration into the element. First, a composite model of cracks

was developed to establish how the cracks participate in the total transport through

the element. From that model, the total permeability was calculated as a function of

the intact concrete properties and the crack parameters. A simple parallel and series

composite model was used, yielding an analytical expression for the total permeability.

From this expression, the influence each crack parameter has on the overall permeability

was investigated and are discussed in the following sections.

3.1 Composite Model

The influence of cracks on the transport properties of a slab with span L and depth h can

be characterized analytically using the schematic shown in Fig. 1. Let there be m cracks,

each with width w, perpendicular to the entire the span, and each crack penetrating into

the slab a proportion α of the total depth. The permeability of the uncracked concrete

is ko. The permeability of each crack kc is based on the assumption that the crack walls

10
are parallel and planar [15]:

kc =

w

2

12

(3)

It is assumed that this equation will be applied to the cracks sufficiently large that a

continuum description of the fluid applies and that molecular effects may be neglected.

The total flow through the slab can be estimated by assuming a parallel and series

model for the materials. First, the permeability of the cracked and uncracked portion up

to the depth αh is calculated. Second, this composite value is put in series with the final

(1 − α)h depth of uncracked concrete to determine the overall composite permeability.

The composite permeability kp of the slab to the depth αh is a weighted sum of the two

permeabilities (two conductors in parallel):

kp =



1 −

mw

L



ko +

mw

L

kc (4)

The bulk permeability kb of the entire slab is estimated by analogy to electrical conductors

in series:

kb =

1 − α

ko

+

α

kp

(5)

This relationship may be simplified, expressing the total bulk permeability kb as a function of only kp and α:

kb =

ko

1 − (1 − ko/kp) α

(6)

To demonstrate the behavior of Eqn. 6, consider a slab containing 100 μm wide

cracks spaced 1 m apart. Each crack penetrates to a fraction α of the total depth h. The

permeability of the uncracked concrete is 10−18 m2. The ratio kb/kp is plotted in Fig. 2

as a function of the fractional depth α. For this general case of moderate crack width,

the ratio of kb/ko increases as (1 − α)

−1

.

3.2 Sensitivity Analysis

The objective of the sensitivity analysis is to express the bulk permeability kb as a

function of the crack parameters: crack width w, number of cracks m over span L, and

crack penetration fraction α. From this expression, one can make a Taylor expansion for

a change ∆kb in the bulk permeability, expressed as a function of changes in the crack

parameters: ∆w, ∆m, and ∆α.

The expression for the bulk permeability kb is expressed as a function of the crack

parameters:

kb = ko



1 + β

1 + (1 − α)β



(7)

The intermediate quantity β is a function of the crack width w, the span L, and the

number of cracks m:

β(w, m) = mw

L



kc

ko

− 1



(8)

11
0.0 0.2 0.4 0.6 0.8 1.0

α

1

10

100

1000

kb /ko

Figure 2: The ratio of the bulk permeability kb to the intact concrete permeability ko

plotted as a function of crack penetration ratio α. The crack parameters are found in

the text.

Given that the permeability ko of concrete is in the range 10−16 m2to 10−18 m2, and that

cracks in concrete have widths on the order of 10−4 m, the parameter β can be simplified

by assuming planar, parallel cracks:

β =

mw3

12koL

(9)

Using the aforementioned example of 10−18 m2 permeability concrete containing 100 μm

cracks spaced 1 m apart, the value of β is approximately 105.

Using an expression for the bulk permeability kb as a function of the crack parameters, changes in the bulk permeability ∆kb can be expressed as a Taylor expansion that

depends, up to first order, on changes in the crack parameters:

∆kb =

dkb

dm ∆m +

dkb

dw ∆w +

dkb

dα ∆α + · · · (10)

12
The complete differentials are evaluated using the chain rule:

dkb

dm =

∂kb

∂β

∂β

∂m = ko



kb

ko

2

αβ

(1+β)

2

1

m

dkb

dw =

∂kb

∂β

∂β

∂w ≈ ko



kb

ko

2

αβ

(1+β)

2

3

w

dkb

dα =

∂kb

∂α = ko



kb

ko

2

αβ

(1+β)

2

1+β

α

(11)

Substitution for the differentials into Eqn. 10 gives the following relationship:

∆kb

kb

=



kb

ko



αβ

1 + β



1

1 + β

∆m

m

+

3

1 + β

∆w

w

+

∆α

α



(12)

The fractional penetration α is equivalent to the ratio of the crack penetration depth δ

to the total depth h. Therefore, the relative change ∆α/α is equivalent to the fraction

change in the depth:

∆kb

kb

=



kb

ko



αβ

1 + β



1

1 + β

∆m

m

+

3

1 + β

∆w

w

+

∆δ

δ



(13)

The effects of changes in the relative number of cracks m and the crack widths w are

reduced by a factor of β (β  1). The ∆δ term, however, lacks β in the denominator.

Therefore, relative changes in the crack depth can be dramatic, as compared to relative

changes in either the crack width or number of cracks.

During condition assessment, efforts should focus on those aspects that are most

influential to the overall performance of the structure. The previous result suggests that

efforts to characterize cracks in existing structures should concentrate on estimating the

depth of crack penetration.

3.3 Crack Parameter Uncertainty

Characterizing the crack parameters during an assessment requires formulating an estimate for both the value of the parameter and its uncertainty. The uncertainties in crack

parameters are important because they can be included in the estimated uncertainty in

the bulk permeability. A concise definition of uncertainty can be found in either the ISO

technical report [16] or the corresponding NIST technical report [17]. For this discussion,

the uncertainty will be characterized by the estimated standard deviation s.

In cases where an analytical expression relates an observable quantity to dependent

variables, the estimated standard deviation can be calculated from partial differentials

and the corresponding uncertainty in the dependent variable [18]. For the case of the

bulk permeability kb, the estimated variance s

2 will be a function of the intact concrete

permeability ko, the number of cracks m, the width of the cracks w, and the relative

crack penetration depth α [18, 19]:

s

2

kb =



dkb

dko

2

s

2

ko +



dkb

dm2

s

2

m +



dkb

dw 2

s

2

w +



dkb

dα 2

s

2

α + · · · (14)

13
The last three differentials can be taken from Eqn. 11 above. The first term in Eqn. 14

may be evaluated using the chain rule:

dkb

dko

=

∂kb

∂ko

+

∂kb

∂β

∂β

∂ko

= ko



kb

ko

2 

ko

kb

−

αβ

(1 + β)

2



1

ko

(15)

With all the differential quantities calculated, Eqn. 14 may be rewritten succinctly as a

function of the various coefficients of variation (standard deviation divided by value):



skb

kb

2

=



kb

ko

2 

αβ

(1 + β)

2

2

(16)

×

"

ko

kb

(1 + β)

2

αβ − 1

2 

sko

ko

2

+

sm

m

2

+ 9 sw

w

2

+ (1 + β)

2

sα

α

2

#

As for the sensitivity analysis, the coefficient of variation for α is equivalent to the

coefficient of variation in the measured crack penetration depth δ.

The result in Eqn. 16 again demonstrates the need for careful characterization of the

crack penetration depth. As in the sensitivity analysis, the contribution of the relative

uncertainty in the penetration fraction α is a factor of (1 + β) greater than the contribution from the relative uncertainty in the other crack parameters. Therefore, to achieve

similar magnitudes in the relative contribution from each crack parameter, the relative

uncertainty in the crack penetration ratio α would have to be a factor of (1 + β) smaller

than the others.

3.4 Extensive Cracking

In some cases, cracking could be sufficiently severe that the performance of the engineering system cannot be ensured in the existing condition. Fortunately, sealants can be used

to penetrate and fill cracks within the concrete. Eliminating the paths of rapid transport

within the concrete will dramatically reduce the overall permeability of the structure.

A possible deficiency of synthetic sealants is their unknown long term (over centuries)

performance. In the absence of demonstrable proof of the long term performance of the

sealant, a suitable monitoring and remediation program for the sealant could be used to

gain confidence in the estimated long-term performance.

3.5 Subsequent Crack Initiation

While it is important to characterize existing cracks in the performance assessment prior

to entombment, it is also important to enumerate possible causes of subsequent cracking. It has been established that cracks are the primary means of advective (Darcy)

14
transport through concrete. Therefore, subsequent cracking could decrease the expected

performance of a concrete structure.

Subsequent cracking within the concrete can arise from two main causes: mechanical

loading, and chemical attack. Mechanically induced cracks could arise from the physical

process of entombment as the contents of the structure will be moved around and the

volume of the structure filled with grout. The presence of chlorides or sulfates in the

groundwater could eventually initiate chemical reactions, leading to internal stresses and,

ultimately, cracking.

3.5.1 Mechanically induced cracking

During entombment, a number of construction and demolition processes may induce

some measure of added cracking within the concrete structure. These processes may

occur during the removal of a portion of the contents or during the possible in-filling of

the remaining structure. They include both mechanically and thermally induced cracks.

After spent fuel has been removed from the site, and the entombed components returned

to the structure, the interior may be grouted with a cementitious material. The remaining entombed structures may be a few stories tall, and cementitious grouts are typically

two, or more, times as dense as water. If grouted in a single lift, the hydrostatic forces

would be substantial, as would the hoop stresses exerted on the structure. In addition, a

cementitious grout would likely involve an exothermic reaction that could generate thermal gradients in the existing concrete structure. Therefore, the grout material selection,

placement strategies, and construction sequence must be considered simultaneously.

3.5.2 Chemically induced cracking

Even though an assessment at the time of entombment may reveal no evidence of chemical attack, changes in the environment may occur for a number of reasons. One must

consider any environmental changes or new construction that can alter the ground water

composition in contact with the concrete. Specifically, sulfate (SO2−

4

) can react with aluminate phases to form ettringite. In addition, divalent ionic species such as magnesium

(Mg2+) can substitute for calcium within the CSH structure, forming a compound with

dramatically reduced mechanical properties.

3.6 Other Major Transport Pathways

In addition to the presence of cracks, there are other possible major transport pathways

through the structure. These pathways could include construction joints, sealed joints,

conduits, etc. The properties of joints could be characterized, to some degree, by suitable

NDE techniques. In addition, NDE could be used to search for unknown pathways such

as segregation or casting voids.

15
4 Sampling Plan

An accurate characterization of the required geometrical and material properties can

be obtained only with an adequate sampling plan. Although the material properties

can be estimated using existing numerical and computer models, only direct observation

can guarantee that the values used in the assessment truly reflect the condition of the

structure. Moreover, a proper sampling plan is required to reduce the uncertainty in the

material parameters to acceptable levels.

While statistics such as the mean can characterize the central portion of the material

parameter distribution, a well-defined means of characterizing the “tail” of the distribution is needed. Generally, if material property values near the 50-th percentile result

in estimated failure, the design would be summarily rejected. The expected scenario

consists of analyzing cases in which material properties above the 90-th percentile lead

to failure. Therefore, one must carefully characterize the “tail” of the material property

distribution. The estimated shape of the “tail” will depend both on the underlying distribution for the material property and on the number of samples taken to estimate the

parameters of the distribution.

Although the true distribution in the population of material property values may not

be known, one could assume an analytical distribution and then test whether the distribution can characterize the data. The two most likely candidates for distributions are the

normal (Gaussian) and the lognormal distributions. (For a lognormal distribution, the

logarithms are normally distributed.) Both distributions lend themselves to established

tests of normality for hypothesis testing. If the data can be shown to originate from

a particular distribution, the “tails” of the distribution can be estimated with greater

confidence.

4.1 Stratified Random Sampling

A stratified random sampling plan is a means of obtaining information about both the

uncertainty in a material property and any spatial dependence that might exist. Figure 3

shows a schematic of the sampling plan. In the figure, a structural element is divided

into subsections, represented by dashed lines. Samples, represented by filled circles in

the figure, are chosen at random locations within each subsection.

Naturally, not every concrete element is orthorhombic. Therefore, in practice one

must make engineering judgments regarding the best way to subdivide a particular element. Furthermore, restrictions on the allowable number of samples will further complicate the process of subdividing the structure. Nonetheless, a sampling plan that follows

the philosophy of distributing the samples from various parts of the entire structure will

have the best chance of accurately characterizing the condition of the concrete.

4.2 Probabilistic Calculation Requirements

To formulate statistical service lifetime estimates, there has been a universal change

from deterministic to probabilistic performance assessment calculations. Deterministic

16
Figure 3: Schematic of a stratified random sampling plan. A structural element is subdivided (dashed lines) into subsections. Samples (filled circles) are taken from random

locations within each subsection.

calculations (a single calculation based on the “best” input parameter values) are limited

in their ability to formulate an uncertainty statement. These probabilistic calculations are

composed of repeated deterministic calculations where the input parameters are chosen

at random from a statistical distribution. The resulting values form a distribution of

service lifetimes. The statistical, regulatory, and engineering advantages of probabilistic

calculations are significant. The results are rational, defensible statements about the

expected performance of a structure.

Unfortunately, relatively little discussion has been made about the commensurate

sampling requirements. Probabilistic performance assessment software can conduct an

arbitrary number of Monte Carlo calculations based on parameter uncertainty. Increasing

the number of Monte Carlo iterations NMC exposes the “tails” of the service life distribution. In cases where the analytical expression for the output distribution is unknown

(virtually all the time), the minimum characterizable percentile of the output distribution is proportional to 1/NMC. Therefore, as one desires information about smaller and

smaller percentiles of the service life distribution, one simply increases the number of

Monte Carlo calculations.

This approach has an important consequence. As the number of Monte Carlo iterations NMC increases to better characterize the limits of the output distribution, one must

be sure that the “tails” of the input parameters are suitably characterized. Although

the measured values of the parameters are characterized by a probability distribution,

17
there is uncertainty in the distribution parameters (confidence interval) due to the finite

sample size. Distribution parameters based on a relatively small number of samples could

have relatively large uncertainties, resulting in a significant change in the location of the

“tail” percentiles.

4.3 Sample Size Effect

Assuming that a suitable distribution for the material property values has been found,

the distribution parameters will be estimated from a number of specimens taken from

a structure. For the normal distribution, the parameters are the mean and standard

deviation. Because these parameters are estimated from a finite sample size, there is an

estimable level of uncertainty in these parameters. These uncertainties should then be

reflected in the final Monte Carlo calculation.

In principle, the uncertainty in the parameters can be introduced into the Monte Carlo

method in a straightforward way. As an example, consider the normal distribution, the

parameters of which are the mean and standard deviation. To perform the Monte Carlo

calculation, one begins with a normal N(0, 1) random deviate Z and calculates a material

property random deviate Y :

Y = μ + σZ (17)

where μ and σ are the true mean and standard deviation, respectively. Unfortunately,

because the total population is greater than the sample population, the true mean and

standard deviation are unknowable, and can only be estimated from the mean x and

standard deviation s calculated from the n individual measurements xi

:

x =

1

n

Xn

i=1

xi s =

1

n − 1

Xn

i=1

(xi − x)

2

(18)

To calculate a random material deviate, one uses these estimated parameters:

Y = x + sZ (19)

There is, however, uncertainty in the estimated parameters, and the true mean and

standard deviation might differ significantly from the estimated values. Fortunately, the

relationship between the estimated and true parameters can themselves be characterized

by probability distributions.

4.3.1 Chi-square distribution

When the standard deviation of a sample is estimated from a finite number of observations, the estimated standard deviation s generally differs from the true standard deviation σ. For a sample of n observations from which the standard deviation was calculated,

let v represent a ratio of estimated to true values:

v =

n s2

σ

2

(20)

18
0

5 10 15 20

v

0.0

0.2

0.4

0.6

qk

k=1

k=2

k=3

k=4

k=5

k=10

Figure 4: Chi-square distribution gk(v) for different degrees of freedom k.

Values of this ratio v obey a Chi-square distribution gk with (k = n − 1) degrees of

freedom:

gk(v) = v

k/2−1

e

−v/2

2

k/2 Γ(k/2) (21)

The distribution gk is shown in Fig. 4 for different values of k. For small values of

k, the most probable outcome is to underestimate the true standard deviation. As the

number of observations increases, the most probable outcome is to accurately estimate

the true standard deviation.

Using a Chi-square random deviate V with k degrees of freedom, a randomized standard deviation σ

0

that incorporates the uncertainty due to finite sample size can be

calculated from the sample standard deviation s:

σ

0 =

s

p

V/n

(22)

4.3.2 Student t-distribution

When the mean of a population x is estimated from a finite number of observations,

the estimated mean x generally will differ from the true mean μ. For a sample of n

observations from which the arithmetic mean is calculated, let t represent the difference

between the estimated and true means:

t =

x − μ

s/√n

(23)

19
−6 −4 −2

0246

t

0.0

0.1

0.2

0.3

0.4

0.5

hk

k=1

k=2

k=5

Figure 5: Student t-distribution hk(t) for 1, 2, and 5 degrees of freedom k. The dashed

line is the N(0, 1) distribution.

Values of t obey a Student t-distribution hk with (k = n − 1) degrees of freedom:

hk(t) = Γ [(k + 1)/2]

Γ(k/2)√πk 

1 +

t

2

k

−(k+1)/2

(24)

In the limit k → ∞, the hk approaches the normal distribution with mean 0 and variance

1: N(0, 1).

The Student t-distribution hk is shown in Fig. 5 for different values of k; the normal

N(0, 1) is also shown as a dashed line as a reference.

This uncertainty can be built into the estimated mean. Using a Student-t random

deviate T with k degrees of freedom, a randomized mean μ

0

can be calculated from the

estimated mean x and estimated standard deviation s:

μ

0 = x + T

s

√

n

(25)

4.3.3 Adjusted Distribution

The random material deviate Y accounts for finite sample size by combining the Chisquare adjustment to the sample standard deviation s and the Student-t adjustment to

the sample mean x. The final expression starts with the ideal relationship of Eqn. 17

and substitutes the adjusted standard deviation σ

0

(Eqn. 22) and the adjusted mean μ

0

20
(Eqn. 25):

Y = μ

0 + Zσ0

(26)

=



x +

sT

√

n



+

Zs

p

V/n

This expression is a function of the measured quantities x and s, and the random deviates

T (Student-t), Z (normal N(0, 1)), and V (Chi-square), and the number of observations

n.

4.4 Analytical Representation

The effect of finite sample size can be examined analytically for a material parameter

characterized by a normal distribution f(x; μ, σ):

f(x; μ, σ) =

exp h−

1

2



x−μ

σ

2

i

√

2π σ

(27)

The distribution that accounts for the finite sample size is calculated by integrating

over the possible values of μ and σ. These possible values depend on the sample size and

the calculated mean x and standard deviation s. Let pdf(μ; x) and pdf(σ; s) represent

the PDFs of the mean μ and standard deviation s. The final distribution p(y; x, s) is

calculated by integrating over the possible values for μ and σ [20]:

p(y; x, s) = Z

μ,σ

f(y; μ, σ) pdf(μ; x) pdf(σ; s) dμ dσ (28)

Ignoring the effects of finite sample size, and assigning x and s to μ and σ, respectively,

is mathematically equivalent to using the delta function for the respective PDF:

pdf(μ; x) = δ(μ − x) pdf(σ; s) = δ(σ − s) (29)

Substituting these distributions into Eqn. 28 yields the normal distribution:

p(y; x, s) = f(y; x, s) (30)

This is the typical way of generating random material parameters, but does not account

for the finite sample size, and the commensurate uncertainty in the estimated mean and

standard deviation.

Accounting for finite sample size requires the appropriate PDF for μ and σ. The

appropriate distributions are the Student t and Chi-square, respectively:

pdf(μ; x, s) dμ = hk(t)









dt

dμ









dμ pdf(σ; s) dσ = gk(v)









dv

dμ









dμ

= hk



x − μ

s/√n

 







−

√

n

s









dμ = gk



n

s

2

σ

2

 







2ns2

σ

3









dσ

21
Substituting into Eqn. 28 yields the complete expression for the integral:

p(y; x, s) = Z

μ,σ

f(y; μ, σ) hk



x − μ

s/√n



gk



n

s

2

σ

2

 2n

3/2

s

σ

3



dμ dσ (31)

02468 10

y

0.0

0.1

0.2

0.3

0.4

p(y;0,1)

3

1

2

4

5

10

Figure 6: Modified material parameter distribution to account for finite sample size. The

degrees of freedom k are labeled along the ordinate. The dashed line denotes the N(0,1)

distribution.

As a demonstration, the distribution p(y; x = 0, s = 1) is plotted in Fig. 6 for different

degrees of freedom. Because the function p(y; x, s) is symmetric about y = 0, it is plotted

only for positive values of y to show detail. Also in Fig. 6 is the normal N(0, 1) distribution, shown as the dashed line. As the number of observations increases, the output

function approaches that of the normal distribution. For small numbers of observations,

however, the distribution is considerably wider than the normal distribution.

4.5 95-th Quantile

The effect of sample size on the probability of obtaining large values can be substantial.

Consider, as an example, the 95-th quantile y.95 of p(y; x, s):

0.95 = Z y.95

−∞

p(y; x, s) dy (32)

Therefore, 5 % of the random values will be larger than y.95.

The values of y.95 for varying degrees of freedom k are shown in Table 1, and are

plotted as a function of k

−1

in Fig. 7. Also shown in the table are the corresponding

Student-t statistic t.95 and the ratio of the Chi-square statistic to the degrees of freedom

22
0.0 0.2 0.4 0.6 0.8 1.0

k

-1

0.0

2.0

4.0

6.0

8.0

10.0

12.0

y.95

Figure 7: y.95 as a function of the degrees of freedom k.

χ

2

.95/k. The value of y.95 is greater than either of the other statistics. Therefore, using

either statistic by itself would not fully capture the effect of sample size.

In effect, the ratio y.95/1.65 is the multiplier by which the 95-th quantile changes with

respect to k. For k = ∞, the value of y.95 is equal to the 95-th quantile of the normal

distribution. At k = 1, the value of y.95 has increased by a factor of seven.

23
Table 1: The 95-th quantile y.95 for the distribution p(y; x, s). Also shown are the Studentt statistic t0.95 and the ratio of the Chi-square statistic to the degrees of freedom χ

2

.95/k.

k y.95 t.95 χ

2

.95/k

1 11.29 6.31 3.84

2 3.67 2.92 3.00

3 2.73 2.35 2.60

4 2.40 2.13 2.37

5 2.20 2.02 2.21

10 1.90 1.81 1.83

20 1.77 1.73 1.57

∞ 1.65 1.65 1.00

24
5 Example

To demonstrate the assessment method, a fictitious scenario will be developed and analyzed in this Section. The service life of the element will be based on the initiation

of corrosion in the internal steel elements. The corrosion model is considered because

the sulfate model is empirical, giving only the rate of sulfate erosion, and failure due to

leaching is an extremely long term failure mechanism.

100 cm

45 cm

Figure 8: Example scenario showing 100 cm thick wall with 10 cm diameter reinforcement

gallery in the center.

5.1 Scenario

The example concrete element will be a wall that is 1 m thick. In the middle of the

element is a steel post-tensioning gallery. A schematic of the system is shown in Fig. 8.

The element will be exposed to ground water containing sodium chloride. The service life

will be the time required for a sufficient concentration of chlorides to initiate corrosion

25
in the steel gallery. For the 4SIGHT model, the critical chloride concentration is that at

which the ratio of chlorides to hydroxyls ([Cl−]/[OH−]) is greater than 0.6.

5.2 Concrete Mixture

Table 2: Input parameters to the ACI 211 mixture design procedure.

Property Value

Compressive Strength 30 MPa

Maximum Aggregate Size 25 mm

Coarse Aggregate Density 2800 kg/m3

Fine Aggregate Density 2600 kg/m3

Element Type Mass Concrete

Air Entrainment Moderate Exposure

The concrete mixture used in this example was obtained from an ACI 211 mixture

design calculation. The concrete is composed of ordinary portland cement (OPC), with no

supplemental pozzolanic mineral admixtures. The structure is to be exposed to freezing

and thawing, so air entrainment is required. The input parameters to the ACI 211

calculation are given in Table 2.

The output from the ACI 211 mixture design calculation is shown in Table 3. The

ACI 211 design code is only meant as a useful starting point in designing a mixture.

Therefore, the fact that the volume fractions do not sum to one is not a great concern.

They are still useful as a starting point for this example calculation. In practice, the

analyst has the actual mixture design from which to calculate the appropriate volume

fractions. These estimates can be verified independently from a petrographic examination

of a polished concrete section.

Table 3: Concrete mixture design based on ACI 211 for normal strength concrete. The

corresponding w/c is 0.45, and the estimated slump is 51 mm.

Material kg/m3 Volume Fraction

Water 164 0.164

Cement 365 0.114

Coarse Agg. 966 0.345

Fine Agg. 899 0.345

Entrained Air 0.045

26
5.3 Material Parameters

A number of material parameters are required to perform a 4SIGHT calculation. In most

cases, these parameters can be estimated from the mixture design. Direct measurement,

however, is the preferred method.

5.3.1 Capillary Porosity

The capillary porosity φ can be estimated using the water:cement ratio  and the degree

of hydration α [1]:

φ = 1 −

1 + 1.16α

1 + 3.2

= 0.162 (33)

For this calculation, it is assumed that the degree of hydration α is 0.90, which should

be reasonable for a 30 year old concrete structure. This value for the capillary porosity

is used as input to the 4SIGHT calculation, and will be useful in subsequent calculations.

5.3.2 Formation Factor

There are at least two ways to estimate the concrete formation factor. The first is to

use the model of Bentz et al. [21, 22] that requires the water:cement ratio , the degree

of hydration α, and the volume fraction of aggregate (0.690). The calculation can be

performed using the Virtual Cement and Concrete Testing Laboratory (VCCTL) web

site (http://vcctl.cbt.nist.gov).

Strictly speaking, the VCCTL web site returns an estimate for the chloride diffusion

coefficient. This value will be converted to an estimate for the formation factor by using

the self-diffusion coefficient of chloride at 25 ◦C [23] (2.03×10−9 m2/s). For this mixture

design, the estimated chloride diffusion coefficient is 9.3×10−13 m2/s. The corresponding

formation factor is the ratio of the chloride self-diffusion coefficient to this value:

F =

2.03 × 10−9

9.3 × 10−13 = 2183 (34)

This value can be checked for consistency using a model for the formation factor of

a paste and Archie’s law. A model for estimating the formation factor of a paste Fpaste

was developed [13, 24] that is a function of only the capillary porosity φ:

F

−1

paste = 0.001 + 0.07φ

2 + 1.8(φ − 0.18)2H(φ − 0.18)

= 0.00284

= (352)−1

(35)

The function H(x) is the Heaviside function (and equals zero in this case because the

estimated porosity from Eqn. 33 is less than 0.18). The paste formation factor Fpaste

is approximately 352. The volume fraction of paste vpaste within the concrete is 0.265.

27
Based on Archie’s law, the formation factor for the concrete Fconc is a function of the

paste formation factor and the volume fraction of paste:

Fconc ≈ Fpastev

−1.5

paste = 2580 (36)

This value is consistent with the formation factor of 2183 reported by the VCCTL model.

In addition to an estimated diffusion coefficient, the VCCTL model also gives an

interval ([4.7,18.0] ×10−13 ms/s) having a 90 % coverage factor. The corresponding 90 %

coverage for formation factor F.90 can be calculated in a manner similar to Eqn. 34 above:

F.90 =

2.03 × 10−9

[4.7, 18.0] × 10−13 = [1128, 4319] (37)

This information is sufficient to now examine a suitable distribution function to characterize the formation factor uncertainty.

A suitable distribution can be found by considering the midpoint of the F.90 range.

The midpoint of the range is 2723, which differs significantly from the estimated value

2183. Consider, however, the logarithm (base 10) of the values:

log F = 3.34

log 4319 + log 1128

2

= 3.34

Therefore, the logarithms of the range F.95 are symmetric about the logarithm of the

predicted value. We conclude that the formation factor uncertainty is best characterized

by a lognormal distribution. The remaining task is to determine some characteristic

standard deviation.

The standard deviation can be approximated from a comparison between the 90 %

coverage and two standard deviations. Two standard deviations, for a normal distribution, has a coverage of approximately 95 %. The scaling between the 90 % coverage range

F.90 and two standard deviations can be approximated by the ratio of the corresponding

quantiles of the normal distribution:

log F.95 ≈ log F.90

1.960

1.645

(38)

Moreover, an estimate of one standard deviation would be one quarter of this new range:

σ =

[log 4319 + log 1128] 

1.960

1.645 

4

= 0.174 (39)

The final result is that the formation factor can be characterized by the following relationship:

F = 103.34±0.174 (40)

The uncertainty represents one standard deviation, as required for the 4SIGHT input.

28
5.3.3 Permeability

An approximation for the permeability can be estimated from an analytical relationship

[1] that was developed from the review article by Hearn [10]:

k ≈ 105.0−21 m2(41)

Based on the data shown in Figure 20 of Hearn, a coverage of 95 % would span approximately three orders of magnitude. Given this, and given that the uncertainty in the

permeability is best approximated by the lognormal distribution, the permeability can

be easily expressed:

k = 10−18.75±0.75 m2(42)

The uncertainty represents one standard deviation.

5.4 Pore Solution

The composition of the pore solution can be approximated using the work of both Taylor

[14] and Reardon [25]. To simplify the analysis, it is assumed that only the potassium,

sodium, sulfate, and hydroxide concentration are significant. Although the oxide contents

of a cement are easily determined, for the sake of this example, the following oxide mass

fractions are used (see column one of Table 3 in Ref. [25]):

Oxide Mass Fraction

K2O 1.18

Na2O 0.41

These oxide contents can then be combined with Taylor’s model [14] for predicting the

pore solution composition. The analysis is based on a 100 g sample of cement. The

corresponding number of moles of sodium and potassium in the sample are as follows:

Element moles per 100 g cement

K 0.0251

Na 0.0132

According to the Taylor model, the analyst determines the fraction of each cement mineral

species that has reacted. Because it is assumed that the degree of hydration is 0.90, the

calculation is simplified by assuming the quantity of each elemental species released into

the pore solution is equal to the total number of moles present.

The concentrations of the potassium and sodium ionic species is the ratio of the moles

of element released mr to a corrected volume of water. The correction first accounts for

the volume of bound water Vb and then adjusts for binding of species b into the hydration

products:

c(mol/100 g H2O) = mr

100 − Vb + b

The volume of bound water Vb from a fully hydrated 100 g cement sample is approximately 31.6 cm3

[14]. The species binding parameters are the following [14]:

29
Species b (cm3

)

K+ 20

Na+ 31

Upon substitution for these quantities, the final estimates for the concentrations are as

follows:

Species Molality (mol/kgw)

K+ 0.654

Na+ 0.267

According to Taylor, the sulfate concentration can be approximated from the potassium

and sodium concentrations:

mSO2−

4

≈ 0.06 (mK+ + mNa+ )

2 = 0.051mol/kgw

5.5 Portlandite Content

It is assumed that every 100 g of cement (in the absence of supplemental pozzolanic

mineral admixtures) will produce 30 g of portlandite. Based on this assumption, the

concentration of portlandite produced, per kg water, can be approximated from the

water:cement ratio :

mCa(OH)2 ≈

4.05



mol/kgw

For the 0.45 water:cement ratio used here, the expected portlandite concentration is 9.0

mol/kgw.

5.6 Cracks

For the purpose of this example, it will be assumed that cracks are detected in the

concrete element. The cracks are approximately 100 μm wide, spaced 10 m apart, and

extend into the wall to a depth that is approximately 40 % of the entire wall thickness.

The standard deviation of crack thickness is 25 μm, and the uncertainty in the crack

penetration depth is ±20 % of the wall thickness; the standard deviation is assumed to

be one half this, or 0.10. Using the aforementioned information, the ratio of the bulk

permeability (intact concrete plus cracks) to the intact concrete is a factor of 1.67.

5.7 Negative Internal Hydrostatic Pressure

For this example, it will be assumed that the interior of the structure will be maintained

at a negative pressure approximately equal to 10132 Pa (0.1 atm). Analysis will be

performed with both no pressure difference, and the 10.1 kPa difference for comparison

purposes.

30
5.8 Exposure

The exposure condition is assumed to be a brackish groundwater. As a reference, exposure to seawater is also considered. In every case, the exposure concentration is assumed

to be constant over the life of the concrete element.

Table 4: Seawater composition based on Schlesinger (Table 9.1, p. 263). The brackish

water has a salinity of approximately 3.5 g/kg.

Seawater Brackish

Species mg/kg mol/L mol/L

Cl− 19350 0.546 0.0546

Na+ 10760 0.468 0.0468

SO2−

4

2712 0.028 0.0028

Mg2+ 1294 0.053 0.0053

Ca2+ 412 0.010 0.0010

K+ 399 0.010 0.0010

The composition of seawater is fairly constant throughout the world, with the salinity

varying from 30 g/kg to 40 g/kg [26]. On the Atlantic and Pacific coasts of the United

States, the salinity is approximately 35 g/kg and 34 g/kg, respectively (See Reference [26],

Figure 9.3, p. 259).

The particular seawater composition used here is shown in Table 4. Although additional diffusing species are given in the reference, only species appearing at a concentration of at least 0.010 mol/L are included.

0 100 200 300 400

Tf (year)

0.0

0.2

0.4

0.6

0.8

1.0

Pf

Diffusion

Permeation

Permeation w/ Cracks

(a)

0 100 200 300

t (year)

-11

-10

-9

log10

(

K [cm/s])

Diffusion

Permeation

(b)

Figure 9: Response of concrete exposed to seawater: (a) Probability of failure Pf as a

function of the failure time Tf ; and (b) The logarithm of hydraulic conductivity K as a

function of of time t. The error bars represent one standard deviation.

31
5.9 Results

For both the seawater and brackish water exposures, different combinations of diffusive and advective transport are studied; advective transport is achieved by applying a

10.1 kPa negative pressure within the structure’s interior. For the advective transport,

both cracked and uncracked concretes are considered.

For each combination of exposure and transport, the calculation is repeated 50 times

to obtain useful statistics. For each iteration, the time to failure Tf (the initiation of

corrosion) is recorded. These values are sorted in ascending order and plotted against

their ordinal value. The result is a cumulative probability plot, as a function of time t,

for the fraction of systems that failed before time t.

In addition, for each of the 50 iterations, the bulk permeability is calculated periodically. The mean and standard deviation of the bulk permeability at time t are calculated

from the population of 50. This gives, in effect, the time-dependent behavior for the bulk

permeability.

5.9.1 Seawater Results

Exposure to seawater is used as a reference. The probability of failure Pf and the

logarithm of the hydraulic conductivity K as a function of time are shown in Fig. 9 as a

function of time. The results for the case of no negative internal pressure are shown as

the “Diffusion” curve in Fig. 9(a). In the presence of a pressure head, but in the absence

of cracks, the probability of failure, at a give time, increases dramatically, as expected.

Because the effect of the cracks is to increase the permeability by a factor 1.67, the effect

of cracks is small by comparison to the other variables.

The evolution of the bulk hydraulic conductivity K over time, as shown in Fig. 9(b) is

revealing. For each of the 50 iterations, the logarithm of the bulk hydraulic conductivity

was recorded as a function of time. The average logarithm is shown, along with the

standard deviation of the logarithms. Although the error bars overlap, the deviations

are symmetric about the average value.

The interesting feature to note is that, although the duration of the service life decreased with the addition of permeation, the bulk hydraulic conductivity also decreased.

This occurs because, as the portlandite is leached from the concrete, the Darcy flow carries the excess calcium ions deeper into the concrete, where they reprecipitate, decreasing

the porosity. The region within the concrete having the decreased porosity also has a

decreased permeability. In cases with 1-D flow, if any portion of the system has a low

permeability, it will have a dramatic effect on the overall bulk permeability.

5.9.2 Brackish Results

The results for the calculations using the brackish water exposure are shown in Fig. 10.

Because a brackish exposure is more likely than a seawater exposure, a number of additional scenarios were considered.

In Fig. 10(a), the failure probability increases as the systems advance from purely

diffusive D systems, to permeative K systems. The dot-dash system is purely diffusive,

32
0 100 200 300 400

Tf (year)

0.0

0.2

0.4

0.6

0.8

1.0

Pf

D

K

K x 10

K x 10,α x 2

(a)

0 100 200 300 400

t (year)

-11

-10

-9

-8

-7

log10

(

K [cm/s])

D

K

K + Crack

Kx10 + Crack

(b)

0 100 200 300 400

t (year)

-11

-10

-9

-8

-7

log10

(

K [cm/s])

D

K

(c)

0 100 200 300 400

t (year)

-11

-10

-9

-8

-7

log10

(

K [cm/s])

K + Crack

Kx10 + Crack

(d)

Figure 10: Response of concrete exposed to brackish water (salinity: 3.5 g/kg): (a)

Probability of failure Pf as a function of failure time Tf . Permeating systems shown with

solid lines and the addition of cracks denoted by the dashed line in proximity to the solid

line system; and (b) The logarithm of hydraulic conductivity K as a function of of time

t (mean values shown); and (c,d) The logarithm of the hydraulic conductivity K, as in

(b). Error bars represent one standard deviation.

and has the lowest probability of failure. With the addition of a pressure head (solid

curve with filled circles), the failure probability increases. Interestingly, adding cracks

(dashed line) again does not change the results appreciably for the reason mentioned

previously.

Increasing the concrete permeability (in the absence of cracks) by a factor of ten

increased the failure probability considerably. As in the previous case, however, the

addition of cracks (dashed line) did not change the failure probability appreciably.

While the transition from a permeating system to a permeating system with cracks

penetrating 40 % of the concrete thickness did not change the response appreciably, the

extension of the cracks to 80 % of the concrete thickness (dot-dash line) did have a

noticeable effect on the failure probability.

The time dependence of the bulk hydraulic conductivity K, shown in Figs. 10(b-d),

33
exhibits behavior similar to the seawater scenario. The diffusive systems exhibited virtually no change in the bulk hydraulic conductivity. The addition of a Darcy flow, however,

decreased the bulk hydraulic conductivity for the same reasons as for the seawater case.

It should be noted that changes in the bulk permeability can be achieved only through

changes in the total capillary porosity due to salt dissolution and precipitation. Presently,

corresponding changes in the transport coefficients are based solely on changes in the

capillary porosity of the intact concrete and not changes in the transport properties of

the cracks. Additional work is needed to consider transport in the intact concrete and the

cracks separately so that the effects of dissolution/precipitation on the crack transport

properties can be incorporated into changes in the transport coefficients.

34
6 Summary

A performance assessment method has been described and demonstrated for cementbased structures considered for entombment. The assessment relies upon careful characterization of both the material properties of the intact concrete and the following crack

properties: crack width, crack spacing, and crack penetration depth. In addition, the

uncertainty in each material property is incorporated into a Monte Carlo calculation

incorporating parameter uncertainty.

The effects of cracks can be extensive. A sensitivity analysis shows that crack width

and relative crack penetration depth have the greatest effect on bulk permeability. Given

the same relative change in the two crack properties, the crack penetration depth has the

greatest effect on the bulk properties. Therefore, the development of nondestructive techniques for characterizing crack properties should concentrate on accurate determination

of the crack penetration depth.

To address uncertainty in the material parameters, guidelines for sampling and for

distribution characterization have been discussed. The proposed sampling strategy considers both the spatial distribution of the sampling and the uncertainty due to a finite

number of samples; the mean comes from the Student t-distribution and the standard

deviation comes from the Chi-Square distribution, both with a number of degrees of

freedom that depends upon the sample size. Analysis shows that very small sample size

can dramatically increase the uncertainty in the calculated distribution parameters. A

measure of this effect was demonstrated by calculating the 95-th percentile of the sample

distribution as a function of the number of degrees of freedom. For the case of only two

samples, the distance between the mean and the 95-th percentile increases by a factor of

seven.

An example assessment was calculated for exposure to a brackish groundwater with

a salinity of 3.5 g/kg (seawater is typically 35 g/kg). The estimated time until the onset

of corrosion was calculated for different transport modes: diffusion; diffusion and permeation through intact concrete; diffusion and permeation through cracked concrete. For

each scenario considered, the calculation was repeated 50 times, each time incorporating

material parameter uncertainty, to yield a statistical statement about the expected lifetime before corrosion initiation. While the addition of cracks to the calculation always

increases the bulk permeability, the changes in the service life were not distinguishable

for cracks penetrating to 40 % of the element thickness.

The long-term behavior of the bulk permeability of diffusion systems remained virtually constant throughout their service life. The permeability of systems undergoing Darcy

flow decreased over time. The decrease was due to the dissolution of calcium hydroxide

at the surface and transport of calcium ions into the concrete. Deeper in the concrete,

the higher pH re-precipitated the calcium hydroxide, reducing the porosity, and thereby

reducing the bulk permeability. Because of this type of effect, the behavior of cracks

during dissolution/precipitation may be very important, and warrants further research.

35
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