Jacques M Bahi
email: jacques.bahi@univ-fcomte.fr

Xiaole Fang
email: xiaole.fang@univ-fcomte.fr

Christophe Guyeux
email: christophe.guyeux@univ-fcomte.fr

State-of-the-art in Chaotic Iterations based pseudorandom numbers generators Application in Information Hiding

Keywords: Internet Security, Pseudorandom Number Generators, Information Hiding, Discrete Chaotic Iterations

The confidentiality of information transmitted through the Internet requires an intensive use of pseudorandom number generators having strong security properties. For instance, these generators are used to produce encryption keys, to encrypt data with a one-time pad process, or to dissimulate information into cover media. In the previous International Conference on Evolving Internet (Internet 09, 10, and 11), we have proposed the use of discrete chaotic iterations to build pseudorandom number generators that receive two inputted possibly deficient generators, and mix them to produce pseudorandom numbers with high statistical qualities. In this article, we summarize these contributions and we propose simple applications of these generators for encryption and information hiding. For each application, firsts experimental evaluations are given, showing that an attacker using these statistics as detection tools cannot infer the presence of an hidden message into given cover documents.

I. INTRODUCTION

Since pseudorandom sequences are easy to be generated and processed, and due to their need in almost all cryptographic protocols and information hiding schemes, PRNGs are widely used for a secure Internet use. Among other things, they are part of the keys generation of any asymmetric cryptosystem, they produce keystreams in symmetric cryptosystems, they determine which bits will receive the secret message in information hiding, and so on. However, a lot of existing pseudorandom number generators (PRNGs) used in numerical simulations are eliminated for such applications, due to the requirements of speed, statistical quality, and security in that context.

Recent years, some researchers have investigated with success the use of chaotic dynamical systems to generate pseudorandom sequences [START_REF] Hu | A true random number generator based on mouse movement and chaotic cryptography[END_REF], [START_REF] De Micco | Randomizing nonlinear maps via symbolic dynamics[END_REF]. Indeed, chaotic systems have many advantages as unpredictability or disorder-like, which are needed when producing complex sequences. They are extremely sensitive to the initial states too: a minute difference can cause a significant change in output. All these features fit well the requirements of PRNGs, thus explaining the proposal of such dynamics to secure exchanges. For instance, in [START_REF] Larger | Nonlinear dynamics Optoelectronic chaos[END_REF], chaos has been applied to strengthen some optical communications. However, chaotic systems using real numbers on infinite bit representation, realized in finite computing precision, lead to short cycle length, non-ideal distribution, and other deflation of this kind. This is why chaotic systems on a infinite space of integers have been dig for these years, leading to the proposition to use chaotic iterations (CIs) techniques to reach the desired goals [START_REF] Wang | A novel pseudo-random generator based on discrete chaotic iterations for cryptographic applications[END_REF].

Having these goals in mind, we have investigating the proposition to mix secure and fast PRNGs, to take benefits from their respective qualities [START_REF] Bahi | A new chaos-based watermarking algorithm[END_REF], [START_REF] Bahi | Topological chaos and chaotic iterations, application to hash functions[END_REF]. In [START_REF] Bahi | Topological chaos and chaotic iterations, application to hash functions[END_REF], CIs have been proven to be a suitable tool for fast computing iterative algorithms on integers satisfying the topological chaotic property, as it has been defined by Devaney [START_REF] Devaney | An Introduction to Chaotic Dynamical Systems[END_REF]. The way that mix two given generators by using these chaotic iterations have been firstly presented in Internet 2009 [START_REF] Wang | A novel pseudo-random generator based on discrete chaotic iterations for cryptographic applications[END_REF]. It was called "Old CIPRNG". Then, further investigations have been proposed in [START_REF] Bahi | A pseudo random numbers generator based on chaotic iterations. application to watermarking[END_REF], [START_REF] Bahi | Topological chaos and chaotic iterations, application to hash functions[END_REF], [START_REF] Bahi | Improving random number generators by chaotic iterations. application in data hiding[END_REF]. These generators were chaotic and able to pass the most stringent batteries of tests, even if the inputted PRNGs were defective. This claim has been verified experimentally, by evaluating the scores of the logistic map, XORshift, and ISAAC generators through these batteries, when considering them alone or after chaotic iterations. Then, in [START_REF] Wang | Randomness quality of CI chaotic generators. application to internet security[END_REF], a new version of this family has been expressed. This so-called "New CIPRNG" family uses a decimation of strategies leading to the improvement of both speed and statistical qualities. Finally, most recently, efficient implementations on GPU (Graphics Processing Unit) using a last family named Xor CIPRNG, have been designed in [START_REF] Bahi | Efficient and cryptographically secure generation of chaotic pseudorandom numbers on gpu[END_REF], showing that a very large quantity of pseudorandom numbers can be generated per second (about 20 Gsamples/s).

The objective of this article is to make a state-of-the-art of chaotic iterations based PRNGs, and to propose a possible use of them in the field of secrecy preservation through the Internet, by using information hiding techniques. Random binary sequences will be generated by the three methods mentioned above and a XORshift generator. The application of these pseudorandom bits for information hiding will be carried out systematically, and results will be discussed in order to verify that an attacker, who has only access to some elementary statistical tests, cannot determine whether hidden information are embedded into cover documents or not.

The remainder of this paper is organized in the following way. In Section II, some basic definitions concerning chaotic iterations and XORshift are recalled. Their use to produce three new families of generators is recalled in the next section. Section IV contains the proposed applications of the PRNGs for information hiding. This summary of our previous works in the field of PRNGs and their applications ends by a conclusion section, where our contribution is summarized and intended future work is presented. II. BASIC REMAINDERS In this section, notations used in this document are introduced, chaotic iterations embedded in the proposed pseudorandom number generators (PRNGs) are defined, and the wellknown XORshift generator is recalled.

A. Notations

S n → the n th term of a sequence S = (S 1 , S 2 , . . .)

v i → the i th component of a vector v = (v 1 , . . . , v n ) f k → k th composition of a function f a; b → the interval {a, a + 1, . . . b} of integers X N → the set of sequences belonging into X strategy → a sequence of 1; N N S → the set of all strategies C k n → the binomial coefficient n k = n! k!(n-k)! ⊕
→ bitwise exclusive or and → the usual shift operators B. Chaotic iterations Definition 1 The set B denoting {0, 1}, let f : B N -→ B N be an "iteration" function and S ∈ S be a chaotic strategy. Then, the so-called chaotic iterations are defined by x 0 ∈ B N , and

∀n ∈ N * , ∀i ∈ 1; N , x n i = x n-1 i if S n = i f (x n-1 ) S n if S n = i.
In other words, at the n th iteration, only the S n -th cell is "iterated".

C. XORshift

XORshift is a category of very fast PRNGs designed by George Marsaglia [START_REF] Marsaglia | Xorshift rngs[END_REF]. It repeatedly uses the transform of exclusive or (XOR) on a number with a bit shifted version of it. The state of a XORshift generator is a vector of bits. At each step, the next state is obtained by applying a given number of XORshift operations to w-bit blocks in the current state, where w = 32 or 64. A XORshift operation is defined as follows. Replace the w-bit block by a bitwise XOR of the original block, with a shifted copy of itself by a positions either to the right or to the left, where 0 < a < w. This Algorithm 1 has a period of 2 32 -1 = 4.29 × 10 9 .

III. CHAOTIC ITERATIONS APPLIED TO PRNGS

In this section, we describe the CIPRNG implementation techniques that can improve the statistical properties of a large variety of defective generators. They all are based on chaotic iterations (CIs), which have been defined in the previous section.

Input: the internal state z (a 32-bit word) Output: y (a 32-bit word) Let N = 4. Some chaotic iterations are fulfilled to generate a sequence (x n ) n∈N ∈ B 4 N of Boolean vectors: the successive states of the iterated system. Some of these vectors are randomly extracted and their components constitute our pseudorandom bit flow [START_REF] Wang | A novel pseudo-random generator based on discrete chaotic iterations for cryptographic applications[END_REF]. Chaotic iterations are realized as follows. Initial state x 0 ∈ B 4 is a Boolean vector taken as a seed and chaotic strategy (S n ) n∈N ∈ 1, 4 N is constructed with P RN G 2 . Lastly, iterate function f is the vectorial Boolean negation. At each iteration, only the S nth component of state x n is updated. Finally, some x n are selected by a sequence m n , provided by a second generator P RN G 1 , as the pseudorandom bit sequence of our generator.

1: z ← z ⊕ (z 13); 2: z ← z ⊕ (z 17); 3: z ← z ⊕ (z 5
The basic design procedure of the Old CI generator is summed up in Algorithm 2. The internal state is x, the output array is r. a and b are those computed by P RN G 1 and P RN G 2 .

Input: the internal state x (an array of 4-bit words) Output: an array r of 4-bit words 1: a ← P RN G 1 (); 2: m ← a mod 2 + 13; 3: while i = 0, . . . , m do behavior of CIs is exploited in order to obtain an unpredictable PRNG constituted by two logistic maps. This novel generator has successfully passed the NIST [START_REF] Rukhin | A statistical test suite for the validation of random number generators and pseudo random number generators for cryptographic applications[END_REF]. Then, in [START_REF] Bahi | A pseudo random numbers generator based on chaotic iterations. application to watermarking[END_REF], we have achieved to improve the speed of the former PRNG, by using two XORshifts in place of the logistic map. In addition, this new version of our PRNG is able to pass the famous DieHARD statistical battery of tests [START_REF] Marsaglia | Diehard: a battery of tests of randomness[END_REF]. Its security has been improved compared to XORshift alone, and to our former PRNG. However, this latter cannot pass the TestU01 [START_REF] Simard | Testu01: A software library in ansi c for empirical testing of random number generators. software users guide[END_REF] battery, widely considered as the most comprehensive and stringent battery of tests. This goal is achieved by using XORshift and ISAAC as P RN G 1 and P RN G 2 in [START_REF] Bahi | Improving random number generators by chaotic iterations. application in data hiding[END_REF].

B. New CIPRNG

The New CI generator is designed by the following process [START_REF] Wang | Randomness quality of CI chaotic generators. application to internet security[END_REF]. First of all, some chaotic iterations have to be done to generate a sequence (x n ) n∈N ∈ B 32 N of Boolean vectors, which are the successive states of the iterated system. Some of these vectors will be randomly extracted and our pseudorandom bit flow will be constituted by their components. Such chaotic iterations are realized as follows. Initial state x 0 ∈ B 32 is a Boolean vector taken as a seed and chaotic strategy (S n ) n∈N ∈ 1, 32 N is an irregular decimation of P RN G 2 sequence, as described in Algorithm 3.

Another time, at each iteration, only the S n -th component of state x n is updated, as follows:

x n i = x n-1 i if i = S n , else x n i = x n-1 i
. Finally, some x n are selected by a sequence m n as the pseudorandom bit sequence of our generator. (m n ) n∈N ∈ M N is computed from P RN G 1 , where M ⊂ N * is a finite nonempty set of integers.

The basic design procedure of the New CI generator is summarized in Algorithm 3. The internal state is x, the output state is r. a and b are those computed by the two input PRNGs. Lastly, the value g 1 (a) is an integer defined as in Eq. 1.

m n = g 1 (y n ) =              0 if 0 y n < C 0 32 , 1 if C 0 32 y n < 1 i=0 C i 32 , 2 if 1 i=0 C i 32 y n < 2 i=0 C i 32 , . . . . . . N if N -1 i=0 C i 32 y n < 1.
(1) This New CI method presented at Internet 2010 has been published in [START_REF] Wang | Randomness quality of CI chaotic generators. application to internet security[END_REF]. It was initially using two XORshifts, showing better speed and statistical performance while preserving chaotic properties of the Old CIPRNG. For more information, the reader is referred to [START_REF] Wang | Randomness quality of CI chaotic generators. application to internet security[END_REF].

C. Xor CIPRNG

Instead of updating only one cell at each iteration as Old CI and New CI, we can try to choose a subset of components and to update them together. Such an attempt leads to a kind of merger of the two random sequences. When the updating function is the vectorial negation, this algorithm can be simply rewritten as follows [START_REF] Bahi | Efficient and cryptographically secure generation of chaotic pseudorandom numbers on gpu[END_REF]:

x 0 ∈ 0, 2 N -1 , S ∈ 0, 2 N -1 N ∀n ∈ N * , x n = x n-1 ⊕ S n , (2)
The single basic component presented in Eq. 2 is of ordinary use as a good elementary brick in various PRNGs. It corresponds to the discrete dynamical system in chaotic iterations.

IV. APPLICATION EVALUATION

In this section, the application of PRNGs using CI methods for information hiding is given.

A. The Proposed Information Hiding Method

Suppose that the size of the image is M × N . The steps of the proposed information hiding algorithm using the CIPRNG family are summed up below. 1) Generate a pseudorandom sequence S of length M ×N using the above CI methods respectively. 2) Transform the image into a M × N integer sequence.

3) The LSBs (Least Significant Bits) of the image integer sequence are replaced by the generated random bits S. These random LSBs will be treated as a keystream. 4) The information (text or picture) to hide is transformed into a binary sequence. 5) The binary message is hiding into the random LSBs of the image sequence, by using the bitwise exclusive or operation between the two sequences, starting from a selected position acting as part of the secret key.

Pseudorandom sequences generated by the three CI methods mentioned in the previous section, with two XORshift generators and a given image, are used in this application to process to an evaluation of the scheme.

B. First Experimental Evaluation of the Proposed Scheme

1) The context: The original image of size 713 × 713, probably the most widely used test image for all kind of processing algorithms (such as compression and encryption), is depicted in Fig. 1-a. Fig. 1-c presents its histogram, and Fig. 1-d shows the correlation distribution of two horizontal adjacent pixels in this original image. Finally, information that must be hidden into it is the picture of Fig. 1-b, which has 89 × 89 pixels.

2) Histogram and Horizontal Correlation: Two XORshift generators are used to generate a random sequence based on the old CI method. Results are shown in Fig. 4. Histograms and correlation distributions (Fig. b,c,d) are very closed to each other, leading to the assumption that such a method can well protect the hidden information when facing statistical attacks. The same experimental validation has been applied to the New CI method using two XORshift generators. Such experiments lead to results that are shown in Fig. 5. These first results are encouraging and confirm that simple histogram and correlation evaluations cannot detect the presence of hidden messages. The same conclusion can be claimed when using the Xor CI generator, as it is depicted in Fig. 7.

3) All directions correlation coefficients analysis: Using an identical experimental evaluation than in [START_REF] Chen | A symmetric image encryption scheme based on 3d chaotic cat maps[END_REF], the correlation coefficients of the horizontal, vertical, and diagonal directions of all the concerned images (original, with random as LSBs, and with secret information in these LSBs) are shown in Table I. It can be experimentally deduced that the correlation properties of these images are very similar to each other. So an attacker, whose intention is to analyze these coefficients in order to detect possible information hiding, cannot attain his/her goal by such a simple experiment. 4) Initial condition sensitivity: One of the most important properties of the chaotic sequences is that they are very sensitive to their initial conditions. This property can help to face an attacker who has access to the whole algorithm and to an approximation of the secret key. His/her intention, in this attack scenario, is to find the exact secret key (the seed of the keystream and the position of the message), by making small changes on this key. If the keystream and the position do not change a lot when the key is slightly updated, then the attacker can converge by small changes to the used secret key. In the experiments of Figure 8, we slightly alter the keys and try to extract the hidden information from the image. We can conclude that such optimistic attempts always fail in recovering the message.

C. A small evaluation of Encryption

The dissimulation has been obtained in this paper by using the CIPRNGs recalled previously as stream cyphers: encryption is the result of the use of the bitwise exclusive or (XOR) between the given message and pseudorandom sequences generated from various CIPRNGs. We can wonder whether an attacker, who has access to the histogram of LSBs, can infer what can of CIPRNG has been used as keystream. For obvious reasons, these histograms should at least be uniform for each PRNG.

For illustration purpose, Lena has been encrypted by such method using each of the three kind of CIPRNGs, and histogram and correlation distribution of the encrypted image have been computed. The resulting images are depicted in Fig. 4 when using the Old CI method, in Fig. 5 for the New CI one, and in Fig. 7 for the last PRNG recalled here. We can show that this first reasonable requirement seems to be respected, even if this illustration is not a proof.

V. CONCLUSION

We have summarizes in this paper our previous contributions in the field of pseudorandom generators, and we have proposed simple illustrative examples of use for information hiding. The three family of CIPRNGs recalled here are namely the Old CI, the New CI, and the Xor CI PRNGs. For each generator, firsts experimental evaluations of a simple information hiding scheme have been realized, to illustrate that that an attacker using simple statistics cannot determine easily, only by regarding the form of histograms or correlation distributions, the presence of an hidden message into a given document. No evidence of dissimulation appears at first glance, when comparing histograms, correlation distribution, or all directions' correlation coefficients. Furthermore, experiments have illustrated high sensitivity to the secret parameters. These simple evaluations do not imply the security of the proposed scheme, they only illustrate that the use of the recalled PRNGs for information hiding can be further investigated by more stringent tools as steganalyzers and mathematical proofs.

Figure 1 :

 1 Figure 1: (a) The original image. (b) Histogram of the original image. (c) Correlation distribution of the original image. (d) The hidden image.

  ); 4: y ← z; 5: return y; Algorithm 1: An arbitrary round of XORshift algorithm A. The Old CIPRNG

4 :Figure 2 :Figure 3 :

 423 Figure 2: (a) Histogram of pixel values when LSBs are replaced by Old CI. (b) Histogram of pixel values when LBSs are an hidden message xored with Old CI. (c) Correlation distribution of two adjacent pixels in Fig.(a). (d) Correlation distribution of two adjacent pixels in Fig.(b).

Figure 4 :Figure 5 : 16 :Algorithm 3 :

 45163 Figure 4: (a) The encrypted Lena (one-time pad using Old CI). (b) Histogram of Fig.(a). (c) Correlation distribution of two adjacent pixels in Fig.(a)

Figure 6 :Figure 7 :

 67 Figure 6: (a) The encrypted Lena (one-time pad using Xor CI). (b) Histogram of Fig.(a). (c) Correlation distribution of two adjacent pixels in Fig.(a)

Figure 8 :

 8 Figure 8: (a) The difference of two random LSBs image using Old CI PRNG with slight change in initial condition. (b) The difference of two random LSBs image using New CI PRNG with slight change in initial condition (c) The difference of two random LSBs image using Xor CI PRNG with slight change in initial condition

Table I :

 I Correlation coefficients of two adjacent pixels in all directions in the original image, random LSBs images and infomation intergraded random LSBs images

	````````Ì mage	Direction	Horizontal	Vertical	Diagonal
	Original image	0.9793	0.9686	0.9488
	Old CI
	no info		0.9792	0.9686	0.9488
	intergrading info	0.9792	0.9686	0.9488
	New CI
	no info		0.9793	0.9686	0.9488
	intergrading info	0.9793	0.9686	0.9488
	Xor CI
	no info		0.9793	0.9686	0.9487
	intergrading info	0.9793	0.9686	0.9487


Laurent Charpin
email: author.laurent.charpin@enpc.fr

Alain Ehrlacher

Simplified model for the transport of Alkali-Silica Reaction gel in concrete porosity

In this work, we focus on estimating the proportion of gel created by Alkali-Silica Reaction (ASR) in concrete that can enter the Interface Transition Zone (ITZ) surrounding an aggregate at a given pressure. We consider the gel as a non-Newtonian fluid following a Darcy law with a threshold pressure gradient while invading a porous medium. We show that at equilibrium, the mass of gel that penetrates in the ITZ can be considered as proportional to the pressure surrounding the aggregates while the ITZ is not full yet, and then constant once the ITZ is full. This is fairly consistent with the common use of a "connected porosity" in the models dealing with mechanical features of ASR to estimate the volume of gel lost in the porosity. We also study briefly the kinetics of this process and the influence of the gel compressibility.

Introduction

As an attempt to understand why, although the gel can penetrate in the porosity of concrete after Alkali-Silica Reaction (ASR), there still exists an important pressure around the aggregate that can lead to cement paste fracture, we wrote a very simple model accounting for the diffusion of an incompressible fluid following Darcy law with a threshold, around an attacked aggregate. This model is motivated by interesting results from [START_REF] Gaboriaud | Structural model of gelation processes of a sodium silicate sol destabilized by calcium ions: combination of SAXS and rheological measurements[END_REF] showing that synthetic gels very close to ASR gels have a threshold fluid behavior in bulk experiments, which might suggest that it is also the case when invading a porous medium. In many models of concrete degradation during ASR, a certain amount of gel is allowed to flow into the cement paste [START_REF] Bažant | Mathematical model for kinetics of alkali-silica reaction in concrete[END_REF][START_REF] Multon | Effects of aggregate size and alkali content on ASR expansion[END_REF][START_REF] Charpin | A computational linear elastic fracture mechanics-based model for alkali-silica reaction[END_REF][START_REF] Charpin | Modèle micromécanique pour l'étude de la réaction alcali-silice[END_REF]. This phenomenon delays the pressure build-up in the porous cement paste surrounding the aggregates, and needs to be taken into account to correctly estimate damage or cracking, for certain kinds of reactive aggregates which induce significant cracking in the cement paste. Our paper aims at studying the invasion of the cement paste by the gel, in order to justify this approach.

Physical set up of the problem

Our goal in this article is to explain why it is relevant, while modeling cracking around aggregates submitted to ASR, to assume that there is a zone that a limited quantity of gel can easily invade. The existence of this buffer zone, which is mainly due to the presence of a high porosity zone around the aggregates called Interface Transition Zone (ITZ), has an important impact on the pressure in the gel-filled cavity, hence on the possibilities of fracture. It is usually taken into account in mechanical models for ASR as a "connected porosity" length [START_REF] Bažant | Mathematical model for kinetics of alkali-silica reaction in concrete[END_REF][START_REF] Multon | Effects of aggregate size and alkali content on ASR expansion[END_REF][START_REF] Charpin | A computational linear elastic fracture mechanics-based model for alkali-silica reaction[END_REF]. To explain this phenomenon, let us first set the equations needed to describe it and the first simplifications we make.

Solid porosity

First we account for the mass conservation of the gel in the porosity. The total porosity of the concrete is named s  . The solid skeleton is assumed infinitely rigid. Therefore, the porosity s  is constant in time. It might however take different values in a thin region surrounding the aggregate (the ITZ) which is of high porosity and can act as a gel reservoir, and the rest of the cement paste, farther from the aggregate.

Gel compressibility

The volume fraction occupied by the gel is

) , ( t x 
. It is such that

s t     ) , ( 0 x
. The gel is a very special material. It is itself a porous material, and the matrix of this porous material also exhibits viscous relaxation [START_REF] Gaboriaud | Structural model of gelation processes of a sodium silicate sol destabilized by calcium ions: combination of SAXS and rheological measurements[END_REF], all these properties evolving with time, as the gel ages and changes chemical environment. Like other porous materials, its stiffness depends on the speed of loading, due to fluid movements inside the gel. Therefore, it is very difficult at the moment to choose an appropriate constitutive behavior for the gel. The main goal of this article is to study the effect of the flow threshold, therefore, we choose a very basic linear elastic law to model the gel elasticity. The gel density writes:

        K t P t ) , ( 1 ) , ( x x   (1)
Where 0  is the density at zero pressure, ) , ( t P x is the pressure written as a function of space x and time t , and K is the bulk modulus of the gel.

Mass conservation

The poromechanics equations are written following the works of [START_REF] Coussy | Poromechanics[END_REF]. The volume flux of the gel is called q . The mass conservation equation therefore writes:

    0 ) , ( ) , ( div ) , ( ) , (     t t t t t x q x x x    (2)

Transport law

For the transport law, our idea is that if it was simply a Darcy law, all pressure gradients would equilibrate and the pressure would remain insufficient to crack the cement paste. We think the gel might have a threshold shear stress below which does not flow, as is reported for various polymeric gels. However, this is a microscopic property and its upscaling to a macro threshold requires a lot of information about the shape and size of the porosity of concrete, and is a very difficult task which requires large computations [START_REF] Sochi | Modelling the flow of yield-stress fluids in porous media[END_REF]. For now we simply assume that this behaviour can be summed up assuming that below a given macroscopic pressure gradient, the gel does not flow [START_REF] Protopas | Two applications of flow in porous media with threshold gradient[END_REF]. Above this gradient, we assume the flux to increase linearly with the pressure gradient. Therefore, we chose the following elementary transport law for the gel: (3)

Where k represents the permeability of the cement paste and 0  the pressure gradient threshold, both different in the ITZ and the bulk cement paste. Both these quantities are mixed properties of the gel and the cement paste. Notice that we here assume that k is independent of the gel saturation of the porous space, which is certainly not the case since there is a vast literature discussing the proper choices for the dependence of permeability on saturation (van Genuchten 1980). However, our main focus here will be on the equilibrium properties of the ingress of the gel in the porous medium, not on the kinetics.

Dimensionless system in the homogeneous spherical case

For now we assume that the cement paste is homogeneous, that is the porosity is constant and equal to s  . The gel appears in a spherical cavity in the cement paste, which we assume to be at constant pressure imp P due to a chemical process creating gel. Here we are interested in the ingress of the gel in the cement paste, which is a phenomenon of relatively short distance compared to the aggregate size. Therefore, we neglect interactions between the various aggregates composing the concrete and assume the cavity where the gel is created is embedded in an infinite cement paste porous medium.

Use of spherical symmetry

Let us start assuming that a spherical cavity is filled with gel under a constant pressure imp P . The surrounding cement pate is assumed homogeneous for now. The first simplification is due to the spherical symmetry. The mass conservation (Eq. 2) rewrites:

    0 ) , ( ) , ( 2 ) , ( ) , ( ) , ( ) , (        r t r q t r r t r q t r t t r t r     (4)
We also assume that due to the fact that all the gel comes from the cavity, the pressure decreases moving away from the cavity. Therefore the transport law (Eq. 3) simplifies to:

If ¶P ¶r > -t 0 , q = 0 If ¶P ¶r < -t 0 , q = -k ¶P ¶r + t 0 é ë ê ù û ú ì í ï ï î ï ï (5)
Which is also represented in Fig. 1.

Figure 1: Transport law

We then take the gel constitutive behaviour into account (Eq. 1) in the mass conservation equation (Eq. 4), and assume that we are in a saturated region, that is the volume fraction occupied by the gel  is equal to the solid porosity

s  : 0 1 2 1                  K P q r r Pq K r q t P K s  (6)
Now we need to replace the volume flux by its expression as a function of the pressure.

We assume for now that we are in the flow regime (that is

0      r P
), and we will check this condition. Using the expression of the flux corresponding to that regime (Eq. 5), we obtain the following differential equation on the pressure:

                                               K P r P k r r P r P K k K P r P K t P s 1 2 1 0 0 2 2    (7)
This differential equation is valid in the saturated region, between the radius p R where the pressure is prescribed equal to imp P , and the end of the saturated region, at radius

) (t R s . For any radius ) (t R r s 
, the concrete is not saturated. Therefore, the pressure is zero, since the gel has enough space. Hence, considering the transport law in Eq. 5, there is no flow. We can then consider that there is a zero pressure imposed at radius

) (t R s
, which is unknown, or consider that this zero pressure is applied at a remote radius  R , and that between

) (t R s and 
R the flux is zero. In any case, it leads to consider that for

) (t R s the gel volume fraction in zero 0  
and that there is a jump to s  when the gel front, which position is

) (t R s
, passes. Finally we get:

                                                 0 0 ) 0 , ( 0 ) , ( ) , ( , 0 0 If 1 0 2 0 1 2 2 , 0 0 If t P p R r P t R P imp P t p R P r P K P r P k r r P r P K k K P r P s K t P r P      (8)

Dimensionless problem

Now let us make this system dimensionless, by setting:

  p s s p p imp p R t R R t r P R P p t R r ) ( ) ( , ) , ( , , , , 0 0                (9)
Where the characteristic time  is yet unknown. Eq. 7 then becomes:

                                                       1 1 2 1 1 2 2 2 kK R s p (10) Where K R p 0   
is a dimensionless parameter comparing the pressure threshold to the gel bulk modulus. We would like the characteristic time not to depend on the gel compressibility, and we know (looking at Eq. 4) that when gel is incompressible, the time dependency disappears of this equation. This leads us to set

0    k R p s 
. Doing so, the full system becomes (writing the main equation in a more compact form):

                                                         0 0 ) 0 , 1 ( 0 ) , ( ) , 1 ( , 0 1 If 1 1 1 , 0 1 If 2 2              p (11)

Equilibrium

It can be studied looking at the equations obtained by making fluxes and time derivatives zero in Eq. 11. Since the evolution of the pressure profile is blocked when we reach the condition ¶P ¶m +1 ³ 0 , the pressure gradient remains such that

1      
. Then the final pressure state is a linear and satisfies the boundary conditions in pressure at

1  
. Therefore in the simple case where the porosity is uniform, the equilibrium dimensionless pressure writes:

P ¥ (m) = p-(m -1) = -(m -m s ¥ ) (12)

Resolution in the incompressible gel case

Non saturated case

If the porosity in a given point is not fully occupied by gel  

s   
, no pressure build-up occurs 0  P . Except close to the interface with the zone which is saturated with fluid, if the vicinity of a point isn't saturated, the pressure is zero close to this point, and so is the pressure gradient. Therefore, owing to the mass conservation (Eq. 4), the volume fraction of gel f doesn't vary. So since the concrete is initially filled with its interstitial solution, f = 0 until the gel front passes, which means that the volume fraction of gel jumps to reach the porosity s  .

Saturated region

The saturated region can also be divided in a region where flow occurs and a region with no flow.

Flowing regime

In this region where

0      r P
, the variation of porosity writes (from 4 and 5):

                     0 2 2 2 ) , (   r P r r P k r t r (13)
Since the gel fills the porosity of the porous medium which is assumed to be undeformable, its volume fraction is constant. that is we have the following ODE on the gel pressure:

0 2 0 2 2               r P r r P (14)
Which can be found in its dimensionless form from Eq. 11 when

0   : 0 1 2 2 2                     (15)
This equation is valid when ¶P ¶m +1 £ 0 , and can be rewritten:

0 1 2                         (16)
Then taking into account the boundary conditions

p   ) , 1 (  and 0 ) ), ( (      s
, we obtain the pressure profile depending on the position of the front (see Fig. 2):

  ) ( 1 1 ) ( 1 1 ) ( 1 ) ( ) , (             s s s s p         (17) Which is valid when 1 ) (   p s   .
Figure 2: Dimensionless pressure profile at a given time

No flow regime

When this condition is reached, which happens everywhere at the same time, since it does not depend on  , the pressure profile is simply linear and consistent with the pressure gradient threshold:

        ) ( ) , ( s (18)
At this point there is no more flow, so

  s s    ) (
, which we find using the first boundary condition:

1    p s 
. The equilibrium pressure profile then writes:

       1 ) , ( p (19)

Determination of the front position

Now we want to study the evolution of the pressure and flux before the gel stops flowing in the porosity. Since the pressure distribution is known explicitly as a function of

) (t R s or ) (  s
in the dimensionless variables, it simply amounts to finding the evolution of this radius. The equation on ) (  s can be found by mass balance. Here we write the mass balance at the front itself. We could also use a global mass balance, equating the mass of gel which has entered up to a given time to the mass of gel present in the porous medium. However, integration difficulties would arise. First, let us write the volume flux at the front with respect to the dimensionless variables defined in (9):

) ( 1 1 ) ( 1 1 ) ( 1 )) ( ( 2 0 0            s s s s p p k k R q                   (20)
Then thinking that when the front passes at a given radius, the volume fraction of gel jumps from zero to s  , and that the speed of the interface is equal to

) ( ' t R s
, this volume flux is also equal to:

s s s p t R R q    ) ( ' )) ( (  (21)
We then end up, equating these two expressions of the volume flux at the front, with a differential equation on the position of the front:

) ( ' ) ( ) ( 1 1 ) ( 1 2 0            s s s s s p p R k     (22) Finally the time constant τ is chosen equal to 0   k R s p
, consistently to the choice made in the general case in section . We then end up with the differential equation:

) ( ) ( ) ( 1 ) ( ' 2         s s s s p     (23)
which is valid at all times except at 0   where 1 ) 0 (     s , which is the initial condition in the integration. This equation can easily be solved numerically, provided that we avoid the singularity at time

0   by choosing ) 0 ( s 
very slightly greater than one. It doesn't change the result at all and allows the equation to be solved easily. We also see that there is only one parameter in this equation:

p . The condition of validity of this equation (flow regime) writes:

1 ) (   p s   (24)
Which is interesting since the limit value of the dimensionless radius if the saturated region is also given by 1

) (   p s  
. That means that since 1 ) 0 (     s , any positive pressure will make at least a little bit of gel enter. Let us plot the dimensionless front position as a function of time (see Fig. 3). Here we chose to apply a dimensionless pressure 1  p , which explains the asymptotic value of the dimensionless front position

) 2 1 (   p
, since as can be seen on Eq. 23 and was mentionned in section 4,

1    p s 
. The evolution is very fast at the beginning (there is a vertical asymptote at 0 



), and it then slows down gently. Once the dimensionless front position has been determined, it is possible to fully reconstruct the pressure field in time, using Eq. 17. We therefore plot the evolution of pressure with time and space (see Fig. 4). We see that the boundary condition 1  p is well respected and that the asymptotic pressure profile is linear as was underlined before (see Eq. 18).

Figure 4: Dimensionless pressure as a function of dimensionless time and radius Therefore, we have a clear idea of how the gel enters the porosity when assumed incompressible. Let us now address the question of the amount of gel at equilibrium.

Equilibrium of fluid injection in an ITZ-surrounded cavity

Pressure distribution

Let us now assume that due to a high porosity zone surrounding the aggregate, which is called ITZ, the threshold becomes non homogeneous. In the ITZ the porosity will be a  , the threshold a  , while in the rest of the cement paste they remain equal to  , s  . Let us study the equilibrium state. We have seen that at equilibrium, the pressure gradient is everywhere equal to the threshold. We define the ratio of the thresholds as ),

e p        1 If ), 1 ( ) (     (27) e e p          1 If ), 1 ( ) 1 ( ) (     (28) And, ) 1 ( 1        e p s ( 29
)
The pressure profile at equilibrium is represented in two cases on Fig. 5.

Injected mass

Let us call

) (t M
the injected mass at time t , and define the dimensionless injected mass as:

0 3 3 4 ) ( ) ( ~    p R M M  (30)
The injected mass is computed by integration of the local density:

        s p s p d R dr r r r M p R R               2 3 0 2 ) , ( 1 ) ( 4 ) ( ) ( 4 ) ( (31)
From this expression, and using Eqs. 25-29, the dimensionless injected mass can be computed. It is useful to take into account that 1  e (the ITZ thickness is much smaller than the aggregate size), and that 1  p (the pressure magnitude is such that the distances reached by the gel in the paste ( 0  imp P if there is no ITZ) are small compared to the aggregate size p R ). Using these two relations, but keeping the dimensionless compressibility parameter η finite, we write the first order expansion of the injected mass in dimensionless pressure and thickness. It writes:

e p p M s a       1 If ), 1 ( 3 ~    (32)  M ¥ = 3j a e(1+hp)+ 3j s (p-le) 1+hp+he(1-l) ( ) , If m s ¥ ³1+ e (33)
The terms proportional to η are all of second order in p and e . However since the bulk modulus of the gel in ASR is not well known, we make no assumption about it. Finally, considering that the porosity of the cement paste s  is much smaller than that of the ITZ, it suggests that it is a good approximation to take the following law for the mass of fluid that escapes the porosity to move into the fluid:

) 1 )( , min( 3 ~p p e M a       (34)
This law expresses that the volume of gel that flows into the cement paste is first proportional to the pressure and then almost constant. If the compressibility of the gel is not neglected, the mass is first quadratic and then linear.

Discussion

Let us now briefly discuss the meaning of our result.

Relation of our result to ASR modelling

Our main result is the expression of the injected mass given in Eq. 34. Let us introduce a chemical time constant c  which represents the characteristic time for the production of the gel (taking into account of the kinetics of the chemical reaction and the transport time of the chemical species to the reaction site). Since ASR is very slow, is seems a good approximation to assume

1  c  
. In this case, we can also consider that we are always at equilibrium and that the pressure profiles labelled ∞ apply at all times. Therefore, the pressure imp P could be computed as the result as a total gel mass balance instead of being imposed, as long as it is an increasing function of time. This is basically what is done in ASR models developed by authors already cited [START_REF] Bažant | Mathematical model for kinetics of alkali-silica reaction in concrete[END_REF][START_REF] Multon | Effects of aggregate size and alkali content on ASR expansion[END_REF] and also the authors of this paper [START_REF] Charpin | A computational linear elastic fracture mechanics-based model for alkali-silica reaction[END_REF][START_REF] Charpin | Modèle micromécanique pour l'étude de la réaction alcali-silice[END_REF]. We see that the most important parameter is the product e a  which gives the total ITZ volume accessible to the gel.

Effect of the parameters

We have introduced various geometrical and mechanical parameters, such as the aggregate size needs to be estimated, in order to be compared to the speed of creation of the products, as mentioned in the previous paragraph.

Limit of the description

ASR is generally considered to induce damage in the aggregates and in the cement paste, depending on the type of aggregates. Of course, our model which assumes spherical symmetry will not be meaningful anymore at the appearance of a crack. However, it would help determining the pressure build-up leading to the appearance of the crack. After the appearance of the crack, the pressure will drop, which we can assume will stop the gel progression in the porous cement paste. In energy-based fracture models as presented by the authors [START_REF] Charpin | A computational linear elastic fracture mechanics-based model for alkali-silica reaction[END_REF][START_REF] Charpin | Modèle micromécanique pour l'étude de la réaction alcali-silice[END_REF] this is of paramount importance. Also, the threshold Darcy law we chose to represent the behaviour if the fluid should be questionned and motivated. There is a lot of work about the upscaling of the properties of fluids from the bulk to flow properties in porous medium [START_REF] Sochi | Modelling the flow of yield-stress fluids in porous media[END_REF]), but we chose not to work on this very complicated issue in order to build a simple model.

Conclusion

In this article we deal with the ideal problem of the ingress of a threshold fluid in the porous medium surrounding a cavity. We address this problem in the case where the fluid is compressible and give the equations which would have to be solved numerically to have the full solution to this problem. We solve for the pressure profile in the case where the fluid is incompressible, which allows us to plot the evolution of the dimensionless front position and dimensionless pressure in this simple case. Finally we show that at equilibrium, when an ITZ is present, it is interesting to consider simple filling laws for the ITZ (mass proportional to the pressure before the ITZ is full and then constant), which is frequently done in mechanical modelling of ASR.

Figure 3 :

 3 Figure 3: Evolution of the dimensionless front position with respect to the dimensionless time

  write in dimensionless variables de expressions of the pressure profile, in the case where

Figure 5 :

 5 Figure 5: Dimensionless pressure profiles at equilibrium for a low pressure such that e p   1

Figure 1 Figure 2 Figure 3

 123 Figure 1

  Figure 4

Acknowledgement

This work was funded by the 'Chaire Durabilité des Structures et des Matériaux pour l'Énergie' at ENPC in France.