Sulfate attack in sewer pipes: Derivation of a concrete corrosion model via two-scale convergence

# Sulfate attack in sewer pipes: Derivation of a concrete corrosion model via two-scale convergence

## Abstract

We explore the homogenization limit and rigorously derive upscaled equations for a microscopic reaction-diffusion system modeling sulfate corrosion in sewer pipes made of concrete. The system, defined in a periodically-perforated domain, is semi-linear, partially dissipative and weakly coupled via a non-linear ordinary differential equation posed on the solid-water interface at the pore level. Firstly, we show the well-posedness of the microscopic model. We then apply homogenization techniques based on two-scale convergence for an uniformly periodic domain and derive upscaled equations together with explicit formulae for the effective diffusion coefficients and reaction constants. We use a boundary unfolding method to pass to the homogenization limit in the non-linear ordinary differential equation. Finally, besides giving its strong formulation, we also prove that the upscaled two-scale model admits a unique solution.

S

ulfate corrosion of concrete, periodic homogenization, semi-linear partially dissipative system, two-scale convergence, periodic unfolding method, multiscale system.

## 1 Introduction

This paper treats the periodic homogenization of a semi-linear reaction-diffusion system coupled with a nonlinear differential equation arising in the modeling of the sulfuric acid attack in sewer pipes made of concrete. The concrete corrosion situation we are dealing with here strongly influences the durability of cement-based materials especially in hot environments leading to spalling of concrete and macroscopic fractures of sewer pipes. It is financially important to have a good estimate on the moment in time when such pipe systems need to be replaced, for instance, at the level of a city like Los Angeles. To get good such practical estimates, one needs on one side easy-to-use macroscopic corrosion models to be used for a numerical forecast of corrosion, while on the other side one needs to ensure the reliability of the averaged models by allowing them to incorporate a certain amount of microstructure information. The relevant question is: How much of this oscillatory-type information is needed to get a sufficiently accurate description of the heterogeneous reality? Due to the complexity of possible shapes of the microstructure, averaging concrete materials is far more difficult than averaging metallic composites with rigorously defined well-packed structure. In this paper, we imagine our concrete piece to be made of a periodically-distributed microstructure. Based on this assumption, we provide here a rigorous justification of the formal asymptotic expansion performed by us (in [1]) for this reaction-diffusion scenario. Note that in [1] upscaled models are derived for a more general situation involving a locally-periodic distribution of perforations1. Locally periodic geometries refer to a special case of -dependent microstructures, where, inherently, the outer normals to (microscopic) inner interfaces are dependent on both spatial slow variable, say , and fast variable, say .

In the framework of this paper, we combine two-scale convergence concepts with the periodic unfolding of interfaces to pass to the homogenization limit (i.e. to , where is a small parameter linked to the relative size of the perforation) for the uniformly periodic case. Here, the outer normals to the inner interfaces are dependent only on the spatial fast variable. For more details on the mathematical modeling of sulfate corrosion of concrete, we refer the reader to [2, 3] (a moving-boundary approach: numerics and formal matched asymptotics), [4] (a two-scale reaction-diffusion system modeling sulfate corrosion), as well as to [5], where a nonlinear Henry-law type transmission condition (modeling transfer across all air-water interfaces present in this sulfatation problem) is analyzed. Mathematical background on periodic homogenization can be found in e.g., [6, 7, 8], while a few relevant (remotely resembling) worked-out examples of this averaging methodology are explained, for instance, in [9, 10, 11, 12, 13, 14]. It is worth noting that, since it deals with the homogenization of a linear Henry-law setting, the paper [11] is related to our approach. The major novelty here compared to [11] is that we now need to pass to the limit in a non-dissipative object, namely a nonlinear ordinary differential equation (ode). The ode is describing sulfatation reaction at the inner water-solid interface – place where corrosion localizes. This aspect makes a rigorous averaging challenging. For instance, compactness-type methods do not work in the case when the nonlinear ode is posed on -dependent surfaces. We circumvent this issue by ”boundary unfolding” the ode. Thus we fix, as independent of , the reaction interface similarly as in [15], and only then we pass to the limit. Alternatively, one could use varifolds (cf. e.g. [16]), since this seems to be the natural framework for the rigorous passage to the limit when both the surface measure and the oscillating sequences depend on . However, we find the boundary unfolding technique easier to adapt to our scenario than the varifolds.

Note that here we approach the corrosion problem deterministically. However, we have reasons to expect that the uniform periodicity assumption can be relaxed by assuming instead a Birkhoff-type ergodicity of the microstructure shapes and positions, and hence, the natural averaging context seems to be the one offered by random fields; see ch. 1, sect. 6 in [17], ch. 8 and 9 in [18], or [19]. But, methodologically, how big is the overlap between homogenizing deterministically locally-periodic distributions of microstructures compared to working in the random fields context? We will treat these and related aspects elsewhere.

The paper is organized as follows: We start off in section 2 (and continue in section 3) with the analysis of the microscopic model. In section 4, we obtain the -independent estimates needed for the passage to the limit . Section 5 contains the main result of the paper: the set of the upscaled two-scal equations.

## 2 The microscopic model

In this section, we describe the geometry of our array of periodic microstructures and briefly indicate the most aggressive chemical reaction mechanism typically active in sewer pipes. Finally, we list the set of microscopic equations.

### 2.1 Basic geometry

Fig. 1 (i) shows a cross-section of a sewer pipe hosting corrosion. We assume that the geometry of the porous medium in question consists of a system of pores periodically distributed inside the three-dimensional cube with and . The exterior boundary of consists of two disjoint, sufficiently smooth parts: - the Neumann boundary and - the Dirichlet boundary. The reference pore, say , has three pairwise disjoint connected domains , and with smooth boundaries and , as shown in Fig. 1 (iii). Moreover, .

Let be a sufficiently small scaling factor denoting the ratio between the characteristic length of the pore and the characteristic length of the domain . Let and be the characteristic functions of the sets and , respectively. The shifted set is defined by

 Ywk:=Y+Σ3j=0kjejfork=(k1,k2,k3)∈Z3,

where is the unit vector. The union of all shifted subsets of multiplied by (and confined within ) defines the perforated domain , namely

 Ωε:=∪k∈Z3{εYwk|εYwk⊂Ω}.

Similarly, , , and denote the union of the shifted subsets (of ) , , and scaled by . Since usually the concrete in sewer pipes is not completely dry, we decide to take into account a partially saturated porous material2. We assume that every pore has three distinct non-overlapping parts: a solid part (grain) which is placed in the center of the pore, the water film which surrounds the solid part, and an air layer bounding the water film and filling the space of as shown in Fig. 1. The air connects neighboring pores to one another. The geometry defined above satisfies the following assumptions:

1. Neither solid nor water-filled parts touch the boundary of the pore.

2. All internal (air-water and water-solid) interfaces are sufficiently smooth and do not touch each other.

These geometrical restrictions imply that the pores are connected by air-filled parts only which is needed not only to give a meaning to functions defined across interfaces, but also to introduce the concept of extension as given, for instance, in [20]. Furthermore, there are no solid-air interfaces.

### 2.2 Description of the chemistry

There are many variants of severe attack to concrete in sewer pipes, we focus here on the most aggressive one – the sulfuric acid attack. The situation can be described briefly as follows: (The anaerobic bacteria in the flowing waste water release hydrogen sulfide gas () within the air space of the pipe. These bacteria are especially active in hot environments. From the air space inside the pipe, 3 enters the pores of the concrete matrix where it diffuses and then dissolves in the pore water. The aerobic bacteria catalyze some of the into sulfuric acid . molecules can move between air-filled part and water-filled part the water-air interfaces [21]. We model this microscopic interfacial transfer via Henry’s law [22], (see the boundary conditions at in (3) and (4)). being an aggressive acid reacts with the solid matrix4 at the solid-water interface, which is made up of cement, sand, and aggregate, and produces gypsum (i.e. ). Here we restrict our attention to a minimal set of chemical reactions mechanisms as suggested in [2], namely.

 ⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩10H++SO−24+org. matter⟶H2S(aq)+4H2O+oxid. matterH2S(aq)+2O2⟶2H++SO−24H2S(aq)⇌H2S(g)2H2O+H++SO−24+CaCO3⟶CaSO4⋅2H2O+HCO−3 (1)

We assume that reactions (1) do not interfere with the mechanics of the solid part of the pores. This is a rather strong assumption since it is known that (1) can actually produce local ruptures of the solid matrix [23]. For more details on the involved cement chemistry and connections to acid corrosion, we refer the reader to [24] (for a nice enumeration of the involved physicochemical mechanisms), [23] (standard textbook on cement chemistry), as well as to [25, 26, 27] and references cited therein. For a mathematical approach of a similar theme related to the conservation and restoration of historical monuments, we refer to the work by R. Natalini and co-workers (cf. e.g. [28]).

### 2.3 Setting of the equations

The data and unknown are given by

 uε10 :Ω⟶R+ - initial % concentration of H2SO4(aq) uε20 :Ω⟶R+ - initial % concentration of H2S(aq) uε30 :Ω⟶R+ - initial % concentration of H2S(g) uε40 :Ω⟶R+ - initial % concentration of moisture uε50 :Ω⟶R+ - initial % concentration of gypsum uD3 :ΓD×(0,T)⟶R+ - % exterior concentration (Dirichlet data) of H2S(g) uε1 :Ωε×(0,T)⟶R -% concentration of H2SO4(aq) uε2 :Ωε×(0,T)⟶R -% concentration of H2S(aq) uε3 :Ωε1×(0,T)⟶R - concentration of H2S(g) uε4 :Ωε×(0,T)⟶R -% concentration of moisture uε5 :Γswε×(0,T)⟶R - concentration of gypsum

All concentrations are viewed as mass concentrations. We consider the following system of mass-balance equations defined at the pore level. The mass-balance equation for is

 ∂tuε1+div(−dε1∇uε1) =−kε1uε1+kε2uε2,x∈Ωε,t∈(0,T) (2) uε1(x,0) =uε10(x),x∈Ωε −nε⋅dε1∇uε1 =0,x∈Γwaε,t∈(0,T) −nε⋅dε1∇uε1 =εη(uε1,uε5),x∈Γswεt∈(0,T).

The mass-balance equation for is given by

 ∂tuε2+div(−dε2∇uε2) =kε1uε1−kε2uε2,x∈Ωε,t∈(0,T), (3) uε2(x,0) =uε20(x),x∈Ωε −nε⋅dε2∇uε2 =ε(aε(x)uε3−bε(x)uε2),x∈Γwaε,t∈(0,T) −nε⋅dε2∇uε2 =0,x∈Γswε,t∈(0,T).

 ∂tuε3+div(−dε3∇uε3) =0,x∈Ωε1,t∈(0,T) (4) uε3(x,0) =uε30(x),x∈Ωε1 −nε⋅dε3∇uε3 =0,x∈ΓN,t∈(0,T) uε3(x,t) =uD3(x,t),x∈ΓD,t∈(0,T) −nε⋅dε3∇uε3 =−ε(aε(x)uε3−bε(x)uε2),x∈Γwaε,t∈(0,T).

The mass-balance equation for moisture follows

 ∂tuε4+div(−dε4∇uε4) =kε1uε1,x∈Ωε,t∈(0,T) (5) uε4(x,0) =uε40(x),x∈Ωε −nε⋅dε4∇uε4 =0,x∈Γwaε,t∈(0,T) −nε⋅dε4∇uε4 =0,x∈Γswε,t∈(0,T).

The mass-balance equation for the gypsum produced at the water-solid interface is

 ∂tuε5 =η(uε1,uε5),x∈Γswε,t∈(0,T) (6) uε5(x,0) =uε50(x),x∈Γswε,t∈(0,T).

## 3 Weak formulation and basic results

We begin this section with a list of notations and function spaces. Then we indicate our working assumptions and give the weak formulation of the microscopic problem; we bring reader’s attention to the well-posedness of the system (2)–(6).

### 3.1 Notations and function spaces

We use , . , and denote the dual pairing of and , the norm in , and the norm in respectively. and will point out the positive and respectively the negative part of the function . We denote by , , and , the space of infinitely differentiable functions in that are periodic of period , the completion of with respect to norm, and the respective quotient space, respectively. Furthermore, . The Sobolev space as a completion of is a Hilbert space equipped with a norm

 ∥φ∥Hβ(Ω)=∥φ∥H[β](Ω)+(∫Ω∫Ω|φ(x)−φ(y)|2|x−y|n+2(β−[β])dxdy)12

and (cf. Theorem 7.57 in [29]) the embedding is continuous. Since we deal with an evolution problem, we need typical Bochner spaces like , , , and . In the analysis of the microscopic model, we use frequently the following trace inequality for dependent hypersurfaces : For , there exists a constant , which is independent of , such that

 ε|φε|2L2(Γε)≤C∗(|φε|2L2(Ωε)+ε2|∇φε|2L2(Ωε)). (7)

The proof of (7) is given in Lemma 3 of [30]. For a function with , the inequality (7) refines into

 ε|φε|2L2(Γε)≤C∗0(|φε|2L2(Ωε)+ε2β∫εΩ∫εΩ|φε(x)−φε(y)|2|x−y|n+2βdxdy), (8)

where is again a constant independent of . For proof of (8), see [15]. To simplify the writing of some of the estimates, we employ the next set of notations:

 di :=min[0,T]×¯Ω∣dεi∣,i∈{1,2,3,4},~di:=min[0,T]×¯Ω∣~dεi∣, Dm :=max[0,T]×Ωε|∂tdεm|,m∈{1,2,3},kj:=min[0,T]×¯Ω∣kεj∣,j∈{1,2} Kj :=min[0,T]×¯Ω∣∂tkεj∣,~kj:=min[0,T]×¯Ω∣~kεj∣, k∞m :=sup(0,T)×Ω∣kεm∣,~k∞m:=sup(0,T)×Ω∣~kεm∣, K∞m :=sup(0,T)×Ω∣∂tkεm∣,Mi:=sup(0,T)×Ω∣uεi∣,i∈{1,2,3,4,5},
 A∞ :=sup(0,T)×Γwaε|aε|,B∞:=sup(0,T)×Γwaε|bε|, A∞ :=sup(0,T)×Γwaε|∂taε|,B∞:=sup(0,T)×Γwaε|∂tbε|, ~a∞ :=sup(0,T)×Γwa|~a|,~b∞:=sup(0,T)×Γwa|~b|, Q∞ :=sups∈(0,T)×Γswε|Q(s)|,¯η:=||η||∞,^η:=||∂tη||∞.

### 3.2 Assumptions on the data and parameters

We consider the following restriction on the data and parameters:

1. , , , for , for every , , .

2. is measurable w.r.t. and and , is sub-linear and locally Lipschitz function and is bounded and locally Lipschitz function such that

 Unknown environment '%

Additionally to (A2), we sometimes assume (A2)’, that is

3. .

4. , , .

5. , , .

6. in .

7. , and are bounded.

8. and for any and .

The assumptions (A1)–(A3), (A5), and (A6) are of technical nature. The first equality in (A4) points out an infinitely fast (equilibrium) Henry law, while the last two equalities remotely resemble a detailed balance in two of the involved chemical reactions.

### 3.3 Weak formulation of the microscopic model

{defn}

Assume (A1) and (A3). We call the vector , a weak solution to (2)–(6) if , such that the following identities hold

 ⟨∂tuε1,φ1⟩(0,T)×Ωε +(d1∇uε1),∇φ1)(0,T)×Ωε (9) =−(k1uε1,φ1)(0,T)×Ωε+(k2uε2,φ1)(0,T)×Ωε −ε(η(uε1,uε4),φ1)(0,T)×Γswε,
 ⟨∂tuε2,φ2⟩(0,T)×Ωε +(dε2∇uε2),∇φ2)(0,T)×Ωε (10) =(kε1uε1,φ2)(0,T)×Ωε−(kε2uε2,φ2)(0,T)×Ωε +ε(aεuε3,φ2)(0,T)×Γwaε−ε(aεuε2,φ2)(0,T)×Γwaε,
 ⟨∂tuε3,φ3⟩(0,T)×Ωε1 =−(dε3∇uε3),∇φ3)(0,T)×Ωε1 (11) −ε(aεuε3,φ3)(0,T)×Γwaε+ε(aεuε2,φ3)(0,T)×Γwaε,
 ⟨∂tuε4,φ4⟩(0,T)×Ωε =−(dε4∇uε4),∇φ4)(0,T)×Ωε+(kε1uε1,φ4)(0,T)×Ωε (12)

for all and together with the ode

 ∂tuε5 = η(uε1,uε5)a.e. on(0,T)×Γwsε (13)

and the initial conditions

 uεi(0,x) =uεi0(x)x∈Ωε for % all i∈{1,2,4}, (14) uε3(0,x) =uε30(x)x∈Ωε1, uε5(0,x) =uε50(x)x∈Γwsε.

### 3.4 Basic results

{lem}

(Positivity and -estimates) Assume (A1)-(A6), and let be arbitrarily chosen. Then the following estimates hold:

• a.e. in , a.e. and a.e. on .

• , , a.e. in , a.e. in and a.e. on .

Proof (i). We test (9)-(12) with element of the space . We obtain the following inequality

 12∂t|uε1−|2+d1|∇uε1−|2 ≤−k1|uε1−|2+k∞2(uε1−,uε2−) (15) −ε(η(uε1,uε5),−uε1−)Γswε.

Note that the first term on the r.h.s of (15) is negative, while the third term is zero because of (A2). We then get

 ∂t|uε1−|2+2d1|∇uε1−|2≤k∞2(|uε1−|2+|uε2−|2). (16)

On the other hand, (10) leads to

 12∂t|uε2−|2+d2|∇uε2−|2 ≤k∞12(|uε1−|2+|uε2−|2) +εa∞(uε2−,uε3−)Γwaε+εb∞|uε2−|2Γwaε.

By the trace inequality (7) (with ), we get

 ∂t|uε2−|2 +2(d2−C∗b∞)|∇uε2−|2≤k∞1(|uε1−|2+|uε2−|2) (17) +2C∗b∞|uε2−|2+2εa∞(uε2−,uε3−)Γwaε.

 ∂t|uε3−|2+2(d3−C∗a∞)|∇uε3−|2≤2εb∞(uε2−,uε3−)Γwaε+2C∗a∞|uε3−|2, (18)

while from (12), we see that

 ∂t|uε4−|2+2d4|∇uε5−|2≤k∞1(|uε1−|2+|uε5−|2). (19)