Sulfate attack in sewer pipes: Derivation of a concrete corrosion model via twoscale convergence
Abstract
We explore the homogenization limit and rigorously derive upscaled equations for a microscopic reactiondiffusion system modeling sulfate corrosion in sewer pipes made of concrete. The system, defined in a periodicallyperforated domain, is semilinear, partially dissipative and weakly coupled via a nonlinear ordinary differential equation posed on the solidwater interface at the pore level. Firstly, we show the wellposedness of the microscopic model. We then apply homogenization techniques based on twoscale 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 nonlinear ordinary differential equation. Finally, besides giving its strong formulation, we also prove that the upscaled twoscale model admits a unique solution.
label1]Tasnim Fatima and label2]Adrian Muntean
ulfate corrosion of concrete, periodic homogenization, semilinear partially dissipative system, twoscale convergence, periodic unfolding method, multiscale system.
1 Introduction
This paper treats the periodic homogenization of a semilinear reactiondiffusion 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 cementbased 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 easytouse 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 oscillatorytype 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 wellpacked structure. In this paper, we imagine our concrete piece to be made of a periodicallydistributed microstructure. Based on this assumption, we provide here a rigorous justification of the formal asymptotic expansion performed by us (in [1]) for this reactiondiffusion scenario. Note that in [1] upscaled models are derived for a more general situation involving a locallyperiodic distribution of perforations^{1}^{1}1The word ”perforation” is seen here as a synonym for ”pore” or ”microstructure”.. 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 twoscale 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 movingboundary approach: numerics and formal matched asymptotics), [4] (a twoscale reactiondiffusion system modeling sulfate corrosion), as well as to [5], where a nonlinear Henrylaw type transmission condition (modeling transfer across all airwater 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) workedout 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 Henrylaw 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 nondissipative object, namely a nonlinear ordinary differential equation (ode). The ode is describing sulfatation reaction at the inner watersolid interface – place where corrosion localizes. This aspect makes a rigorous averaging challenging. For instance, compactnesstype 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 Birkhofftype 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 locallyperiodic 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 twoscal 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 crosssection 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 threedimensional 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
where is the unit vector. The union of all shifted subsets of multiplied by (and confined within ) defines the perforated domain , namely
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 material^{2}^{2}2The solid, water and air parts corresponds to , and , respectively.. We assume that every pore has three distinct nonoverlapping 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:

Neither solid nor waterfilled parts touch the boundary of the pore.

All internal (airwater and watersolid) interfaces are sufficiently smooth and do not touch each other.
These geometrical restrictions imply that the pores are connected by airfilled 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 solidair 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}^{3}3 and refer to gaseous, and respectively, aqueous . 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 airfilled part and waterfilled part the waterair 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 matrix^{4}^{4}4The solid matrix is assumed here to consist of only. This assumption can be removed in the favor of a more complex cement chemistry. at the solidwater 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.
(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 coworkers (cf. e.g. [28]).
2.3 Setting of the equations
The data and unknown are given by
All concentrations are viewed as mass concentrations. We consider the following system of massbalance equations defined at the pore level. The massbalance equation for is
(2)  
The massbalance equation for is given by
(3)  
The massbalance equation for reads
(4)  
The massbalance equation for moisture follows
(5)  
The massbalance equation for the gypsum produced at the watersolid interface is
(6)  
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 wellposedness 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
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
(7) 
The proof of (7) is given in Lemma 3 of [30]. For a function with , the inequality (7) refines into
(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:
3.2 Assumptions on the data and parameters
We consider the following restriction on the data and parameters:

, , , for , for every , , .

is measurable w.r.t. and and , is sublinear and locally Lipschitz function and is bounded and locally Lipschitz function such that
Additionally to (A2), we sometimes assume (A2)’, that is

.

, , .

, , .

in .

, and are bounded.

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}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
(15)  
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
(16) 
On the other hand, (10) leads to
By the trace inequality (7) (with ), we get
(17)  
(11) leads to
(18) 
while from (12), we see that
(19) 
Adding up inequalities (16)(19) gives
(20)  
and hence,
(21)  
Applying the trace inequality (7) to estimate the last term on the right side of (21), we finally get