1 Introduction

Coupled mass transport and linear reaction processes in porous media are widely studied in many engineering domains such as electrochemical systems, oil industry, storage of nuclear waste, environmental and hydro-geological applications and biogeochemistry. In most cases, the porous media are characterized by a multiscale structure requiring the understanding of the multiphysics and multiscale coupling at the pore-scale to develop macroscopic models using upscaling procedures.

Several methods can be used to carry out this upscaling: volume averaging method (Plumb and Whitaker 1988; Valdes-Parada and Alvarez-Ramirez 2010; Valdes-Parada et al. 2011), method of moment (Edwards et al. 1993; Brenner 1980; Shapiro and Brenner 1988), thermodynamically constrained averaging theory Gray and Miller (2014) and periodic homogenization method (Auriault et al. 2009; Auriault and Lewandowska 1996; Bourbatache et al. 2012, 2016, 2012, 2013a, b, 2020; Auriault and Bloch 2019; Moyne and Murad 2006). In these upscaling methods, the information from the local scale is filtered according to space and time scale constraints.

Upscaling diffusion transport coupled to chemical reactions has been widely developed in the literature. In particular, the diffusion-advection-chemical reaction problem has been upscaled using the volume averaging method (Valdes-Parada and Alvarez-Ramirez 2010; Valdes-Parada et al. 2011; Valdés-Parada et al. 2017). The authors have shown that the effective diffusion tensors depend on the chemical reaction rate when the diffusion and reaction effects are of the same order of magnitude. Porta et al. (2013, 2012) have applied the same upscaling method to the transport problem coupled to bimolecular reaction where the effective diffusion tensors are time-dependent function due to the reaction effects. In Guo et al. (2015), the volume averaging method has been used to upscale the mass transport coupled to dissolution reaction at the interface. Lugo-Méndez et al. (2015) have extended the works of Valdes-Parada and Alvarez-Ramirez (2010); Valdés-Parada et al. (Oct 2017) to the case of a nonlinear reaction term and predominant reaction. In this situation, the effective diffusion tensor is shown to be a function of the reaction rate. Qiu et al. (2017) have performed binary species diffusion-advection problem coupled to heterogeneous reaction at the solid–fluid interface. The authors underline the dependency of the effective diffusion tensor on the reaction rate.

Using classical periodic homogenization procedure for diffusion/reaction problems, more complex homogenization processes have been proposed in the literature (Mauri 1991; Allaire and Raphael 2007) introducing auxiliary eigenvalue problems to capture very short times. The authors in Battiato and Tartakovsky (2011) have studied the dispersive transport of single component with linear heterogeneous reaction and concluded that the asymptotic development is applicable only for small Damköhler number. To overcome this difficulty for high Damköhler number, a novel numerical method, namely hybrid method, has been proposed in Battiato et al. (2011), Yousefzadeh and Battiato (2017). This latter couples the solution of the macroscopic equation with the one of the pore-scale model in the domain where high Damköhler number occurs.

In this paper, we propose to test the classical homogenization procedure in the case of coupled diffusion-reaction equations for large Damköhler number values. Three different situations are considered increasing progressively the order of magnitude of the Damköhler number. The first case, rather classical, corresponds to a predominant diffusion where the Damköhler number is small. In this case, a macroscopic model is obtained coupling diffusion and reaction through a source term, where the associated effective diffusion tensors depend only on the geometry (Sect. 4). For a moderate Damköhler number, corresponding to diffusion and reaction of the same order of magnitude, we obtain in Sect. 5 two identical macroscopic equations for the concentrations of the two species with an effective diffusion tensor that depends on the reaction rates when the microscopic diffusion coefficients are different. In this case, it is important to note that the macroscopic system must satisfy a supplementary chemical equilibrium condition at the main order (Eq. (24)). For the case of high Damköhler number (\(\text {Da} = {\mathcal {O}}(\varepsilon ^{-1})\)), namely predominant reaction, a macroscopic model similar to the previous one is obtained with the same chemical equilibrium condition at the main order. Finally, simulations in Sect. 7 reveal that the homogenized models for moderate-high Damköhler numbers are not able to capture the physics at (very) small characteristic time representative of the (very) fast reaction process. Chemical equilibrium is instantaneously reached at the main order, which is inconsistent in most cases with non-equilibrium initial or boundary conditions.

2 Pore-Scale Model

The diffusion mechanism of the two species is governed by a Fickian process in the fluid phase \(\Omega _f^{*}\). The concentrations are denoted by \(c_1^{*}\) and \(c_2^{*}\). At the solid/fluid interface \(\Gamma _{fs}^{*}\), a reversible chemical reaction characterized by the reaction rate coefficients \(k_{1}^{*}\) and \(k_{2}^{*}\) occurs. The microscopic diffusion-reaction equations at the pore-scale are written as

$$\begin{aligned} \left\{ \begin{array}{rcll} \displaystyle \frac{\partial c_1^{*}}{\partial t^{*}} - {\varvec{\nabla }}^{*} \cdot ({\mathcal {D}}_1^{*}{\varvec{\nabla }}^{*} c_1^{*}) &{}=&{} 0 &{} \text{ in } \Omega _{f}^{*} \\ \displaystyle \frac{\partial c_2^{*}}{\partial t^{*}} - {\varvec{\nabla }}^{*} \cdot ({\mathcal {D}}_2^{*}{\varvec{\nabla }}^{*} c_2^{*}) &{}=&{} 0 &{} \text{ in } \Omega _{f}^{*} \\ - {\mathcal {D}}_1^{*}{\varvec{\nabla }}^{*} c_1^{*} \cdot {\mathbf {n}}_{fs} &{}=&{} k_1^{*} c_1^{*} - k_2^{*} c_2^{*} &{} \text{ at } \Gamma _{fs}^{*} \\ -{\mathcal {D}}_2^{*}{\varvec{\nabla }}^{*} c_2^{*} \cdot {\mathbf {n}}_{fs} &{}=&{} k_2^{*} c_2^{*} - k_1^{*} c_1^{*} &{} \text{ at } \Gamma _{fs}^{*} \end{array} \right. \end{aligned}$$
(1)

where \({\mathcal {D}}_1^{*}\) and \({\mathcal {D}}_2^{*}\) denote the diffusion coefficients of the two species and \({\mathbf {n}}_{fs}\) the normal unit vector at the solid/fluid interface pointing out of the fluid phase. The microscopic equations are completed by external boundary and initial conditions giving the pore-scale model.

The macroscopic Damköhler numbers are defined as

$$\begin{aligned} \mathrm {Da}_{1}^{L} = \displaystyle \frac{k_{1}^{*} L}{{\mathcal {D}}_{1}^{*}}; \qquad \mathrm {Da}_{2}^{L} = \displaystyle \frac{k_{2}^{*} L}{{\mathcal {D}}_{2}^{*}} \end{aligned}$$
(2)

We consider that the two Damköhler numbers are of the same order of magnitude, i.e. \({\mathcal {O}}\left( \mathrm {Da}_{1}^{L}\right) = {\mathcal {O}}\left( \mathrm {Da}_{2}^{L}\right) \equiv {\mathcal {O}}\left( \mathrm {Da}^{L}\right)\).

3 Periodic Homogenization Procedure

Our aim is to construct a macroscopic law for the diffusion-reaction problem by upscaling the system of equations (1) using the periodic homogenization procedure Bensoussan et al. (1978); Sanchez Palencia (1980). To do this, a formal analysis based on a dimensional analysis and an asymptotic expansion of the partial differential equations is carried out written with a strong formulation without specifying the functional analysis spacesFootnote 1.

3.1 Periodic Porous Medium

We consider a porous medium occupying a macroscopic domain \(\Omega ^{*}\) with a characteristic length L, composed of an immobile fluid phase \(\Omega _f^{*}\) and of a rigid solid phase \(\Omega _s^{*}\) with a solid–fluid interface \(\Gamma _{fs}^{*}\). The medium \(\Omega ^{*}\) is assumed to be constituted of the repetition of a periodic elementary cell \(Y^{*} = Y^{*}_{f} \cup Y^{*}_{s}\) of characteristic length l, composed of a fluid phase \(Y^*_{f}\) and of a solid phase \(Y^{*}_{s}\). The frontier of the unit cell \(\partial Y^{*}\) with adjacent cells is made of the external fluid (\(\partial Y_{fe}^{*}\)) and solid (\(\partial Y_{se}^{*}\)) boundaries. \(\partial Y^{*}_{fs}\) designates the fluid-solid interface assumed to be impermeable (see Fig. 1). The macroscopic and microscopic spatial coordinates are noted \({\mathbf {x}}^{*}=\left( x^{*}_1,\,x^{*}_2,\, x^{*}_3\right)\) and \({\mathbf {y}}^{*} = \left( y^{*}_1,\, y^{*}_2,\, y^{*}_3\right)\), respectively. The condition of scale separation as \(l \ll L\) assumes that \({\mathbf {x}}^{*}\) and \({\mathbf {y}}^{*}\) are independent variables. Note that \(\varepsilon = l/L\) is a small parameter.

Fig. 1
figure 1

Schematic representation of the porous medium with a periodic microstructure. a Macroscopic scale. b Elementary reference cell

Full size image

3.2 Dimensional Analysis of Equations

First, a dimensional analysis of the equations is performed. Let \(c_r\), \({\mathcal {D}}_r\) and \(k_r\) be the reference quantities for concentrations, diffusion coefficients and reaction rate coefficients, respectively. The dimensionless quantities, which are \({\mathcal {O}}(1)\) quantities, are defined as:

$$\begin{aligned} c_i=\frac{c_i^{*}}{c_{r}},\quad {\mathcal {D}}_{i}=\frac{{\mathcal {D}}_{i}^{*}}{{\mathcal {D}}_{r}},\quad k_{i}= \displaystyle \frac{k_{i}^{*}}{k_{r}} \end{aligned}$$
(3)

for \(i \in \{1,2\}.\)

The reference length scale is chosen as the macroscopic length L, so that we have:

$$\begin{aligned} {{\varvec{\nabla }}}= L \, {\varvec{\nabla }}^{*}, \quad {{\varvec{\nabla }}}\, \cdot = L \, {\varvec{\nabla }}^{*} \, \cdot \end{aligned}$$
(4)

The reference time is chosen as the macroscopic diffusive time corresponding to the observation time:

$$\begin{aligned} t_{r}=\frac{L^2}{{\mathcal {D}}_{r}} \end{aligned}$$
(5)

Finally, the reference macroscopic Damköhler number is defined as

$$\begin{aligned} {\mathrm {Da}}_{r}^{L} = \displaystyle \frac{k_{r} L}{{\mathcal {D}}_{r}} \end{aligned}$$
(6)

Therefore, the dimensional analysis of diffusion-reaction equations (1) leads to

$$\begin{aligned} \left\{ \begin{array}{rcll} \displaystyle \frac{\partial c_1}{\partial t} - {{\varvec{\nabla }}}\cdot ({\mathcal {D}}_1{{\varvec{\nabla }}}c_1) &{}=&{} 0 &{} \text{ in } \Omega _{f} \\ \displaystyle \frac{\partial c_2}{\partial t} - {{\varvec{\nabla }}}\cdot ({\mathcal {D}}_2{{\varvec{\nabla }}}c_2) &{}=&{} 0 &{} \text{ in } \Omega _{f} \\ -{\mathcal {D}}_1{{\varvec{\nabla }}}c_1 \cdot {\mathbf {n}}_{fs} &{}=&{} \mathrm{Da}_r^L( k_1 c_1 - k_2 c_2 ) &{} \text{ at } \Gamma _{fs} \\ -{\mathcal {D}}_2{{\varvec{\nabla }}}c_2 \cdot {\mathbf {n}}_{fs} &{}=&{} \mathrm{Da}_r^L( k_2 c_2 - k_1 c_1) &{} \text{ at } \Gamma _{fs} \end{array} \right. \end{aligned}$$
(7)

3.3 Reduction to One-Scale Problem

In the framework of periodic homogenization technique, the dimensional problem is reduced to a one-scale problem with the scale ratio \(\varepsilon = l/L\) defined as the perturbation parameter. Three different cases are considered in this work, increasing progressively the Damköhler number:

  • Dominant diffusion for \(\mathrm{Da}_{1}^{L}\) and \(\mathrm{Da}_{2}^{L} = {\mathcal {O}}(\varepsilon )\) (\(\mathrm{Da}_{r}^{L}\) is chosen as \(\mathrm{Da}_{r}^{L}=\varepsilon\))

  • Diffusion and reaction of the same order for for \(\mathrm{Da}_{1}^{L}\) and \(\mathrm{Da}_{2}^{L} = {\mathcal {O}}(\varepsilon ^{0})\) (\(\mathrm{Da}_{r}^{L}=\varepsilon ^{0}\))

  • Predominant reaction for \(\mathrm{Da}_{1}^{L}\) and \(\mathrm{Da}_{2}^{L} = {\mathcal {O}}(\varepsilon ^{-1})\) (\(\mathrm{Da}_{r}^{L}=\varepsilon ^{-1}\))

3.4 Asymptotic Expansion

In a classical way, the concentrations admit an asymptotic expansion with respect to the perturbation parameter \(\varepsilon\):

$$\begin{aligned} c_1^{(\varepsilon )}({\mathbf {x}},{\mathbf {y}},t)=\sum _{k=0}^{\infty }\varepsilon ^{k}c_1^{(k)}({\mathbf {x}}, {\mathbf {y}}, t) \ , \qquad c_2^{(\varepsilon )}({\mathbf {x}},{\mathbf {y}},t)=\sum _{k=0}^{\infty }\varepsilon ^{k}c_2^{(k)}({\mathbf {x}}, {\mathbf {y}}, t) \end{aligned}$$
(8)

with the functions \(c_1^{(k)}({\mathbf {x}}, {\mathbf {y}},t)\) and \(c_2^{(k)}({\mathbf {x}}, {\mathbf {y}},t)\) assumed to be \({\mathbf {y}}\)-periodic.

Since the reference length scale is the macroscopic characteristic length L, the differential operators are written as Bensoussan et al. (1978), Sanchez Palencia (1980):

$$\begin{aligned} \begin{array}{rclll} {{\varvec{\nabla }}}f^{(\varepsilon )}({\mathbf {x}}, {\mathbf {y}}, t) &{}=&{} {{\varvec{\nabla }}}_x f^{(\varepsilon )} ({\mathbf {x}},{\mathbf {y}}, t) + \varepsilon ^{-1} {{\varvec{\nabla }}}_y f^{(\varepsilon )}({\mathbf {x}}, {\mathbf {y}},t) \\ {{\varvec{\nabla }}}\cdot f^{(\varepsilon )}({\mathbf {x}}, {\mathbf {y}},t) &{}=&{} {{\varvec{\nabla }}}_x \cdot f^{(\varepsilon )} ({\mathbf {x}},{\mathbf {y}},t) + \varepsilon ^{-1} {{\varvec{\nabla }}}_y \cdot f^{(\varepsilon )}({\mathbf {x}}, {\mathbf {y}},t) \end{array} \end{aligned}$$
(9)

for any function \(f^{(\varepsilon )}\).

Finally, the superficial average and intrinsic average operators in the unit cell are defined as

$$\begin{aligned} \langle f \rangle = \displaystyle \frac{1}{\mid Y \mid } \int _{Y_f} f \mathrm{d}V; \qquad \langle f \rangle ^f = \displaystyle \frac{1}{\mid Y_f \mid } \int _{Y_f} f \mathrm{d}V \end{aligned}$$
(10)

4 Dominant Diffusion

The first case considered corresponds to a predominant diffusion for which \(\text {Da}^{L} = {\mathcal {O}}(\varepsilon )\). In this case, the \(\varepsilon\)-microscopic model to homogenize is given byFootnote 2:

$$\begin{aligned} \left\{ \begin{array}{rcll} \displaystyle \frac{\partial c_1^{(\varepsilon )}}{\partial t} - {{\varvec{\nabla }}}\cdot ({\mathcal {D}}_1{{\varvec{\nabla }}}c_1^{(\varepsilon )}) &{}=&{} 0 &{} \text{ in } Y_{f} \\ \displaystyle \frac{\partial c_2^{(\varepsilon )}}{\partial t} - {{\varvec{\nabla }}}\cdot ({\mathcal {D}}_2{{\varvec{\nabla }}}c_2^{(\varepsilon )}) &{}=&{} 0 &{} \text{ in } Y_{f} \\ -{\mathcal {D}}_1{{\varvec{\nabla }}}c_1^{(\varepsilon )} \cdot {\mathbf {n}}_{fs} &{}=&{} \varepsilon (k_1 c_1^{(\varepsilon )} - k_2 c_2^{(\varepsilon )}) &{} \text{ at } \partial Y_{fs} \\ -{\mathcal {D}}_2{{\varvec{\nabla }}}c_2^{(\varepsilon )} \cdot {\mathbf {n}}_{fs} &{}=&{} \varepsilon (k_2 c_2^{(\varepsilon )} - k_1 c_1^{(\varepsilon )}) &{} \text{ at } \partial Y_{fs} \end{array} \right. \end{aligned}$$
(11)

We then have the following result:

Result 1

For a predominant diffusion corresponding to \(\mathrm{Da}^L = {\mathcal {O}}(\varepsilon )\), at the leading order \(c_{1}^{(0)}\) and \(c_{2}^{(0)}\) are slow variables and solutions of the following macroscopic problem

$$\begin{aligned} \left\{ \begin{array}{llll} \displaystyle \frac{\partial c_1^{(0)}}{\partial t} - {{\varvec{\nabla }}}_x \cdot \left( {\mathbf {D}}_1^{\mathrm{eff}} \cdot {{\varvec{\nabla }}}_x c_1^{(0)} \right) + \displaystyle \frac{|\partial Y_{fs}|}{|Y_f|} \left( k_1 c_1^{(0)} - k_2 c_2^{(0)} \right) &{}=&{} 0 \\ \displaystyle \frac{\partial c_2^{(0)}}{\partial t} - {{\varvec{\nabla }}}_x \cdot \left( {\mathbf {D}}_2^{\mathrm{eff}} \cdot {{\varvec{\nabla }}}_x c_2^{(0)} \right) + \displaystyle \frac{|\partial Y_{fs}|}{|Y_f|} \left( k_2 c_2^{(0)}- k_1 c_1^{(0)} \right) &{}=&{} 0 \end{array} \right. \end{aligned}$$
(12)

The effective diffusion tensors \({\mathbf {D}}_1^{\mathrm{eff}}\) and \({\mathbf {D}}_2^{\mathrm{eff}}\) are defined as

$$\begin{aligned} \begin{array}{llll} {\mathbf {D}}_1^{\mathrm{eff}} &{}=&{} \left\langle {\mathcal {D}}_1 \left( {\varvec{I}} + \left( {{\varvec{\nabla }}}_y {\varvec{\chi }}\right) ^{{\mathsf {T}}}\right) \right\rangle ^f \\ {\mathbf {D}}_2^{\mathrm{eff}} &{}=&{} \left\langle {\mathcal {D}}_2 \left( {\varvec{I}} + \left( {{\varvec{\nabla }}}_y {\varvec{\chi }}\right) ^{{\mathsf {T}}}\right) \right\rangle ^f \end{array} \end{aligned}$$
(13)

where \({\varvec{I}}\) is the identity tensor of order two and where the superscript “\({\mathsf {T}}\)” denotes the transposition operator. The vector \({\varvec{\chi }}\) is solution of the local closure problem

$$\begin{aligned} \left\{ \begin{array}{rcll} {{\varvec{\Delta }}}_{y}{\varvec{\chi }}&{}=&{} 0 &{} \text {in } Y_{f}\\ \left( {\varvec{I}} + {{\varvec{\nabla }}}_{y}{\varvec{\chi }}\right) \cdot {\mathbf {n}}_{fs} &{}=&{} 0 &{} \text {at } \partial Y_{fs}\\ \langle {\varvec{\chi }}\rangle ^f &{}=&{}0\\ {\varvec{\chi }}\left( {\mathbf {y}}\right) &{}=&{}{\varvec{\chi }}\left( {\mathbf {y}}+l \right) \end{array} \right. \end{aligned}$$
(14)

where \({{\varvec{\Delta }}}_{y}\) denotes the Laplacian with respect to the variable y. This problem is completed by periodic conditions at the external boundary of the unit cell. Henceforth, periodic conditions are imposed for all the local problems in the unit cell and will not be explicitly stated.

Proof

the proof of this result is detailed in what follows. \(\square\)

Asymptotic Expansion of Equations

We now collect the successive powers of \(\varepsilon\) in the system of Eq. (11) by considering the asymptotic expansions of \(c_1^{(\varepsilon )}\) and \(c_2^{(\varepsilon )}\) as well as expansion (9) of the spatial derivative operators.

\(\bullet\) Slow Variables

At order \({\mathcal {O}}(\varepsilon ^{-2})\) in the volume and \({\mathcal {O}}(\varepsilon ^{-1})\) at the interface, we have

$$\begin{aligned} \left\{ \begin{array}{rcll} {{\varvec{\nabla }}}_y \cdot ({\mathcal {D}}_1{{\varvec{\nabla }}}_y c_1^{(0)}) &{}=&{} 0 &{} \text{ in } Y_{f} \\ {{\varvec{\nabla }}}_y \cdot ({\mathcal {D}}_2{{\varvec{\nabla }}}_y c_2^{(0)}) &{}=&{} 0 &{} \text{ in } Y_{f} \\ -{\mathcal {D}}_1 {{\varvec{\nabla }}}_y c_1^{(0)} \cdot {\mathbf {n}}_{fs} &{}=&{} 0&{} \text{ at } \partial Y_{fs} \\ -{\mathcal {D}}_2 {{\varvec{\nabla }}}_y c_2^{(0)} \cdot {\mathbf {n}}_{fs} &{}=&{} 0 &{} \text{ at } \partial Y_{fs} \end{array} \right. \end{aligned}$$
(15)

leading straightforwardly to the solution \(c_{1}^{(0)}({\mathbf {x}}, {\mathbf {y}},t) = c_{1}^{(0)}({\mathbf {x}},t)\) and \(c_2^{(0)}({\mathbf {x}}, {\mathbf {y}},t) = c_{2}^{(0)}( {\mathbf {x}},t)\).

\(\bullet\) Closure Problem

At order \({\mathcal {O}}(\varepsilon ^{-1})\) in the volume and \({\mathcal {O}}(\varepsilon ^{0})\) at the interface, it yields

$$\begin{aligned} \left\{ \begin{array}{rcll} {{\varvec{\nabla }}}_y \cdot \left[ {\mathcal {D}}_1({{\varvec{\nabla }}}_x c_1^{(0)}+{{\varvec{\nabla }}}_y c_1^{(1)})\right] &{}=&{} 0 &{} \text{ in } Y_{f} \\ {{\varvec{\nabla }}}_y \cdot \left[ {\mathcal {D}}_2({{\varvec{\nabla }}}_x c_2^{(0)}+{{\varvec{\nabla }}}_y c_2^{(1)})\right] &{}=&{} 0 &{} \text{ in } Y_{f} \\ -{\mathcal {D}}_1({{\varvec{\nabla }}}_x c_1^{(0)}+{{\varvec{\nabla }}}_y c_1^{(1)}) \cdot {\mathbf {n}}_{fs} &{}=&{} 0 &{} \text{ at } \partial Y_{fs} \\ -{\mathcal {D}}_2({{\varvec{\nabla }}}_x c_2^{(0)}+{{\varvec{\nabla }}}_y c_2^{(1)}) \cdot {\mathbf {n}}_{fs} &{}=&{} 0 &{} \text{ at } \partial Y_{fs} \end{array} \right. \end{aligned}$$
(16)

The solutions of \(c_1^{(1)}\) and \(c_2^{(1)}\) are sought in the form

$$\begin{aligned} \begin{array}{rlll} c_1^{(1)} &{}=&{} {\varvec{\chi }}_1 \cdot {{\varvec{\nabla }}}_x c_1^{(0)} + \widehat{c_1}(t, {\mathbf {x}}) \\ c_2^{(1)} &{}=&{} {\varvec{\chi }}_2 \cdot {{\varvec{\nabla }}}_x c_2^{(0)} + \widehat{c_2}(t, {\mathbf {x}}) \end{array} \end{aligned}$$
(17)

where \(\widehat{c_1}(t, {\mathbf {x}})\) and \(\widehat{c_2}(t, {\mathbf {x}})\) are constant in the unit cell and \({\varvec{\chi }}_1\) and \({\varvec{\chi }}_2\) are solutions of the closure problem (14) leading to \({\varvec{\chi }}_1={\varvec{\chi }}_2={\varvec{\chi }}\).

\(\bullet\) Macroscopic Mass Conservation Equations

At the order \({\mathcal {O}}(\varepsilon ^{0})\), we have

$$\begin{aligned} \left\{ \begin{array}{llll} &{}&{}\displaystyle \frac{\partial c_1^{(0)}}{\partial t} - {{\varvec{\nabla }}}_x \cdot \left[ {\mathcal {D}}_1({{\varvec{\nabla }}}_x c_1^{(0)}+{{\varvec{\nabla }}}_y c_1^{(1)}) \right] - {{\varvec{\nabla }}}_y \cdot \left[ {\mathcal {D}}_1({{\varvec{\nabla }}}_x c_1^{(1)}+{{\varvec{\nabla }}}_y c_1^{(2)}) \right] = 0 &{} \text{ in } Y_{f} \\ &{}&{}\displaystyle \frac{\partial c_2^{(0)}}{\partial t} - {{\varvec{\nabla }}}_x \cdot \left[ {\mathcal {D}}_2({{\varvec{\nabla }}}_x c_2^{(0)}+{{\varvec{\nabla }}}_y c_2^{(1)}) \right] - {{\varvec{\nabla }}}_y \cdot \left[ {\mathcal {D}}_2({{\varvec{\nabla }}}_x c_2^{(1)}+{{\varvec{\nabla }}}_y c_2^{(2)}) \right] = 0 &{} \text{ in } Y_{f} \\ &{}&{}-{\mathcal {D}}_1({{\varvec{\nabla }}}_x c_1^{(1)}+{{\varvec{\nabla }}}_y c_1^{(2)}) \cdot {\mathbf {n}}_{fs} = k_1 c_1^{(0)} - k_2 c_2^{(0)} &{} \text{ at } \partial Y_{fs} \\ &{}&{}-{\mathcal {D}}_2({{\varvec{\nabla }}}_x c_2^{(1)}+{{\varvec{\nabla }}}_y c_2^{(2)}) \cdot {\mathbf {n}}_{fs} = k_2 c_2^{(0)} - k_1 c_1^{(0)} &{} \text{ at } \partial Y_{fs} \end{array} \right. \end{aligned}$$
(18)

By averaging equations (18a) and (18b) in the fluid phase of the unit cell, using (17), we obtain the macroscopic mass conservation equations (12) where the effective coefficients \({\mathbf {D}}_1^{\mathrm{eff}}\) and \({\mathbf {D}}_2^{\mathrm{eff}}\) are defined by (13).

Going back to dimensional variables, homogenized diffusion equations at the leading order for the concentrations \(c_{1}^{*(0)}\) and \(c_{2}^{*(0)}\) are written as:

Result 2

For a predominant diffusion corresponding to \(\mathrm{Da}^{L} = {\mathcal {O}}(\varepsilon )\), the concentrations at the leading order are solution of the homogenized coupled diffusion-reaction problem:

$$\begin{aligned} \left\{ \begin{array}{rcl} \displaystyle \frac{\partial c_{1}^{*(0)}}{\partial t^*} = {{\varvec{\nabla }}}_{x^{*}}\cdot \left( {\mathbf {D}}_1^{*\mathrm eff} \cdot {{\varvec{\nabla }}}_{x^{*}}c_{1}^{*(0)}\right) - \displaystyle \frac{|\partial Y_{fs}^*|}{|Y_f^*|} \left( k_{1}^*c_{1}^{*(0)} - k_{2}^*c_{2}^{*(0)} \right) \\ &{} &{} \\ \displaystyle \frac{\partial c_{2}^{*(0)}}{\partial t^*} = {{\varvec{\nabla }}}_{x^{*}}\cdot \left( {\mathbf {D}}_2^{*\mathrm eff} \cdot {{\varvec{\nabla }}}_{x^{*}}c_{2}^{*(0)}\right) - \displaystyle \frac{|\partial Y_{fs}^*|}{|Y_f^*|} \left( k_{2}^*c_{2}^{*(0)} - k_{1}^*c_{1}^{*(0)} \right) \end{array} \right. \end{aligned}$$
(19)

The homogenized diffusion tensors are given by

$$\begin{aligned} \begin{array}{rcl} {\mathbf {D}}_1^{*\mathrm eff} &{}=&{} \left\langle {\mathcal {D}}_{1}^*\left( {\varvec{I}} +\left( {{\varvec{\nabla }}}_{y^{*}}{\varvec{\chi }}^*\right) ^{{\mathsf {T}}} \right) \right\rangle ^{f} \\ {\mathbf {D}}_2^{*\mathrm eff} &{}=&{} \left\langle {\mathcal {D}}_{2}^*\left( {\varvec{I}} + \left( {{\varvec{\nabla }}}_{y^{*}}{\varvec{\chi }}^*\right) ^{{\mathsf {T}}} \right) \right\rangle ^{f} \end{array} \end{aligned}$$
(20)

where \({\varvec{\chi }}^*\) is solution of the classical closure problem for diffusion:

$$\begin{aligned} \left\{ \begin{array}{rcll} {{\varvec{\nabla }}}_{y^{*}}\cdot \left( {\varvec{I}} + \left( {{\varvec{\nabla }}}_{y^{*}}{\varvec{\chi }}^*\right) ^{{\mathsf {T}}}\right) &{}=&{} 0 &{} \text {in } Y_{f}^*\\ \left( {\varvec{I}} + {{\varvec{\nabla }}}_{y^{*}}{\varvec{\chi }}^*\right) \cdot {\mathbf {n}}_{fs} &{}=&{} 0 &{} \text {at } \partial Y_{fs}^*\\ \langle {\varvec{\chi }}^*\rangle ^f &{}=&{} 0 \\ {\varvec{\chi }}^*\left( {\mathbf {y}}^*\right) &{}=&{}{\varvec{\chi }}^*\left( {\mathbf {y}}^*+l \right) \end{array} \right. \end{aligned}$$
(21)

5 Diffusion and Reaction of the same Order

The intermediate case corresponds to diffusion and reaction of the same order, i.e satisfying \(\text {Da}^{L} = {\mathcal {O}}(\varepsilon ^{0})\). In that case, the system of equations to homogenize reads as:

$$\begin{aligned} \left\{ \begin{array}{rcll} \displaystyle \frac{\partial c_1^{(\varepsilon )}}{\partial t} - {{\varvec{\nabla }}}\cdot ({\mathcal {D}}_1{{\varvec{\nabla }}}c_1^{(\varepsilon )}) &{}=&{} 0 &{} \text{ in } Y_{f} \\ \displaystyle \frac{\partial c_2^{(\varepsilon )}}{\partial t} - {{\varvec{\nabla }}}\cdot ({\mathcal {D}}_2{{\varvec{\nabla }}}c_2^{(\varepsilon )}) &{}=&{} 0 &{} \text{ in } Y_{f} \\ -{\mathcal {D}}_1{{\varvec{\nabla }}}c_1^{(\varepsilon )} \cdot {\mathbf {n}}_{fs} &{}=&{} k_1 c_1^{(\varepsilon )} - k_2 c_2^{(\varepsilon )} &{} \text{ at } \partial Y_{fs} \\ - {\mathcal {D}}_2{{\varvec{\nabla }}}c_2^{(\varepsilon )} \cdot {\mathbf {n}}_{fs} &{}=&{} k_2 c_2^{(\varepsilon )} - k_1 c_1^{(\varepsilon )} &{} \text{ at } \partial Y_{fs} \end{array} \right. \end{aligned}$$
(22)

leading to the following result.

Result 3

For diffusion and reaction of the same order corresponding to \(\mathrm{Da}^L = {\mathcal {O}}(\varepsilon ^{0})\), at the leading order \(c_{1}^{(0)}\) and \(c_{2}^{(0)}\) are slow variables and solution of the coupled diffusion problem:

$$\begin{aligned} \begin{array}{llll} \displaystyle \frac{\partial c_i^{(0)}}{\partial t} - {{\varvec{\nabla }}}_x \cdot \left( {\mathbf {D}}^{\mathrm{eff}}_{\mathrm{H}} \cdot {{\varvec{\nabla }}}_x c_i^{(0)} \right) &{}=&{} 0 \\ \end{array} \end{aligned}$$
(23)

with \(i=1,2\) together with the constraint

$$\begin{aligned} k_1 c_1^{(0)} - k_2 c_2^{(0)}=0 \end{aligned}$$
(24)

everywhere in the fluid phase. The effective diffusion coefficient \({\mathbf {D}}^{\mathrm{eff}}_{\mathrm{H}}\) is defined by

$$\begin{aligned} {\mathbf {D}}^{\mathrm{eff}}_{\mathrm{H}} = \displaystyle \frac{k_1 {\mathbf {D}}_2^{\mathrm{eff}} + k_2 {\mathbf {D}}_1^{\mathrm{eff}}}{k_1+k_2} \end{aligned}$$
(25)

where the homogenized tensors \({\mathbf {D}}_1^{\mathrm{eff}}\) and \({\mathbf {D}}_2^{\mathrm{eff}}\) are given as

$$\begin{aligned} \begin{array}{llll} {\mathbf {D}}_1^{\mathrm{eff}} &{}=&{} \left\langle {\mathcal {D}}_1\left( {\varvec{I}} + \left( {{\varvec{\nabla }}}_y {\varvec{\chi }}\right) ^{{\mathsf {T}}}\right) \right\rangle ^f \\ {\mathbf {D}}_2^{\mathrm{eff}} &{}=&{} \left\langle {\mathcal {D}}_2\left( {\varvec{I}} + \left( {{\varvec{\nabla }}}_y {\varvec{\chi }}\right) ^{{\mathsf {T}}}\right) \right\rangle ^f \end{array} \end{aligned}$$
(26)

The vector \({\varvec{\chi }}\) is solution of the local problem (14).

Proof

the proof of this result is detailed in what follows.

\(\bullet\) Slow Variables

At order \({\mathcal {O}}(\varepsilon ^{-2})\) in the volume and \({\mathcal {O}}(\varepsilon ^{-1})\) at the interface, we have

$$\begin{aligned} \left\{ \begin{array}{rcll} {{\varvec{\nabla }}}_y \cdot ({\mathcal {D}}_1{{\varvec{\nabla }}}_y c_1^{(0)}) &{}=&{} 0 &{} \text{ in } Y_{f} \\ {{\varvec{\nabla }}}_y \cdot ({\mathcal {D}}_2{{\varvec{\nabla }}}_y c_2^{(0)}) &{}=&{} 0 &{} \text{ in } Y_{f} \\ -{\mathcal {D}}_1 {{\varvec{\nabla }}}_y c_1^{(0)} \cdot {\mathbf {n}}_{fs} &{}=&{} 0 &{} \text{ at } \partial Y_{fs} \\ - {\mathcal {D}}_2 {{\varvec{\nabla }}}_y c_2^{(0)} \cdot {\mathbf {n}}_{fs} &{}=&{} 0 &{} \text{ at } \partial Y_{fs} \end{array} \right. \end{aligned}$$
(27)

whose solution is

$$\begin{aligned} c_1^{(0)}=c_1^{(0)}({\mathbf {x}}, t) ; \qquad c_2^{(0)}=c_2^{(0)}({\mathbf {x}}, t) \end{aligned}$$
(28)

\(\bullet\) Closure Problem

At order \({\mathcal {O}}(\varepsilon ^{-1})\) in the volume and \({\mathcal {O}}(\varepsilon ^{0})\) at the interface, it yields

$$\begin{aligned} \left\{ \begin{array}{rcll} {{\varvec{\nabla }}}_y \cdot \left[ {\mathcal {D}}_1({{\varvec{\nabla }}}_x c_1^{(0)}+{{\varvec{\nabla }}}_y c_1^{(1)})\right] &{}=&{} 0 &{} \text{ in } Y_{f} \\ {{\varvec{\nabla }}}_y \cdot \left[ {\mathcal {D}}_2({{\varvec{\nabla }}}_x c_2^{(0)}+{{\varvec{\nabla }}}_y c_2^{(1)})\right] &{}=&{} 0 &{} \text{ in } Y_{f} \\ -{\mathcal {D}}_1({{\varvec{\nabla }}}_x c_1^{(0)}+{{\varvec{\nabla }}}_y c_1^{(1)}) \cdot {\mathbf {n}}_{fs} &{}=&{} k_1 c_1^{(0)} - k_2 c_2^{(0)} &{} \text{ at } \partial Y_{fs} \\ - {\mathcal {D}}_2({{\varvec{\nabla }}}_x c_2^{(0)}+{{\varvec{\nabla }}}_y c_2^{(1)}) \cdot {\mathbf {n}}_{fs} &{}=&{} k_2 c_2^{(0)} - k_1 c_1^{(0)} &{} \text{ at } \partial Y_{fs} \end{array} \right. \end{aligned}$$
(29)

Averaging (29a) over the unit cell taking into account the interface condition (29c) and the periodicity yields \(\displaystyle { \int _{\partial Y_{fs}} ( k_1 c_1^{(0)} - k_2 c_2^{(0)} ) dy } = 0\) and therefore according to (28), we obtain

$$\begin{aligned} k_1 c_1^{(0)} - k_2 c_2^{(0)} =0 \ \ \text {in} \ Y_f \end{aligned}$$
(30)

The chemical equilibrium is verified everywhere in the fluid phase at the leading order.

According to (30), problem (29) reduces to (16), and therefore, the solutions of \(c_1^{(1)}\) and \(c_2^{(1)}\) can be sought in the form

$$\begin{aligned} \begin{array}{rlll} c_1^{(1)} &{}=&{} {\varvec{\chi }}\cdot {{\varvec{\nabla }}}_x c_1^{(0)} + \widehat{c_1}(t, {\mathbf {x}}) \\ c_2^{(1)} &{}=&{} {\varvec{\chi }}\cdot {{\varvec{\nabla }}}_x c_2^{(0)} + \widehat{c_2}(t, {\mathbf {x}}) \end{array} \end{aligned}$$
(31)

where \(\widehat{c_1}(t, {\mathbf {x}})\) and \(\widehat{c_2}(t, {\mathbf {x}})\) are constant in the unit cell and where the vector \({\varvec{\chi }}\) is the solution of the classical closure problem (14).

\(\bullet\) Macroscopic Mass Conservation Equations

At order \({\mathcal {O}}(\varepsilon ^{0})\), we have

$$\begin{aligned} \left\{ \begin{array}{llll} &{}&{}\displaystyle \frac{\partial c_1^{(0)}}{\partial t} - {{\varvec{\nabla }}}_x \cdot \left[ {\mathcal {D}}_1({{\varvec{\nabla }}}_x c_1^{(0)}+{{\varvec{\nabla }}}_y c_1^{(1)}) \right] - {{\varvec{\nabla }}}_y \cdot \left[ {\mathcal {D}}_1({{\varvec{\nabla }}}_x c_1^{(1)}+{{\varvec{\nabla }}}_y c_1^{(2)}) \right] = 0 &{} \text{ in } Y_{f} \\ &{}&{}\displaystyle \frac{\partial c_2^{(0)}}{\partial t} - {{\varvec{\nabla }}}_x \cdot \left[ {\mathcal {D}}_2({{\varvec{\nabla }}}_x c_2^{(0)}+{{\varvec{\nabla }}}_y c_2^{(1)}) \right] - {{\varvec{\nabla }}}_y \cdot \left[ {\mathcal {D}}_2({{\varvec{\nabla }}}_x c_2^{(1)}+{{\varvec{\nabla }}}_y c_2^{(2)}) \right] = 0 &{} \text{ in } Y_{f} \\ &{}&{}-{\mathcal {D}}_1({{\varvec{\nabla }}}_x c_1^{(1)}+{{\varvec{\nabla }}}_y c_1^{(2)}) \cdot {\mathbf {n}}_{fs} = k_1 c_1^{(1)} - k_2 c_2^{(1)} &{} \text{ at } \partial Y_{fs} \\ &{}&{}- {\mathcal {D}}_2({{\varvec{\nabla }}}_x c_2^{(1)}+{{\varvec{\nabla }}}_y c_2^{(2)}) \cdot {\mathbf {n}}_{fs} = k_2 c_2^{(1)} - k_1 c_1^{(1)} &{} \text{ at } \partial Y_{fs} \end{array} \right. \end{aligned}$$
(32)

It should be noted that

$$\begin{aligned} k_1 c_1^{(1)} - k_2 c_2^{(1)} = {\varvec{\chi }}\cdot {{\varvec{\nabla }}}_x(k_1 c_1^{(0)} - k_2 c_2^{(0)}) + (k_1\widehat{c_1}-k_2\widehat{c_2})= k_1\widehat{c_1}-k_2\widehat{c_2} \end{aligned}$$
(33)

which is an \({\mathbf {x}}\)-dependent variable.

By averaging (32) over the unit cell, we obtain the macroscopic mass conservation equations

$$\begin{aligned} \left\{ \begin{array}{llll} \displaystyle \frac{\partial c_1^{(0)}}{\partial t} - {{\varvec{\nabla }}}_x \cdot \left( {\mathbf {D}}_1^{\mathrm{eff}} \cdot {{\varvec{\nabla }}}_x c_1^{(0)} \right) +f({\mathbf {x}},t) =0\\ \displaystyle \frac{\partial c_2^{(0)}}{\partial t} - {{\varvec{\nabla }}}_x \cdot \left( {\mathbf {D}}_2^{\mathrm{eff}} \cdot {{\varvec{\nabla }}}_x c_2^{(0)} \right) -f({\mathbf {x}},t) = 0 \end{array} \right. \end{aligned}$$
(34)

where \(f({\mathbf {x}},t)\) is given by

$$\begin{aligned} f({\mathbf {x}},t) = \displaystyle \frac{|\partial Y_{fs}|}{|Y_f|} \left( k_1 \widehat{c_1} - k_2 \widehat{c_2}\right) \end{aligned}$$
(35)

together with the constraint \(k_1 c_1^{(0)} - k_2 c_2^{(0)}=0\). The effective coefficients \({\mathbf {D}}_1^{\mathrm{eff}}\) and \({\mathbf {D}}_2^{\mathrm{eff}}\) are given by (26). By using the constraint \(k_1 c_1^{(0)} - k_2 c_2^{(0)}=0\), \(f({\mathbf {x}},t)\) can be determined from (34) as:

$$\begin{aligned} \begin{array}{llll} f({\mathbf {x}},t)= & {} {{\varvec{\nabla }}}_x \cdot \left[ \displaystyle \frac{({\mathbf {D}}_2^{\mathrm{eff}}-{\mathbf {D}}_1^{\mathrm{eff}})k_1}{k_1+k_2} \cdot {{\varvec{\nabla }}}_x c_1^{(0)} \right] \end{array} \end{aligned}$$
(36)

Finally, we have

$$\begin{aligned} \begin{array}{llll} \displaystyle \frac{\partial c_1^{(0)}}{\partial t} - {{\varvec{\nabla }}}_x \cdot \left( {\mathbf {D}}^{\mathrm{eff}}_{\mathrm{H}} \cdot {{\varvec{\nabla }}}_x c_1^{(0)} \right)= & {} 0 \end{array} \end{aligned}$$
(37)

where the effective diffusion coefficient \({\mathbf {D}}^{\mathrm{eff}}_{\mathrm{H}}\) is defined by

$$\begin{aligned} {\mathbf {D}}^{\mathrm{eff}}_{\mathrm{H}} = \displaystyle \frac{k_1 {\mathbf {D}}_2^{\mathrm{eff}} + k_2 {\mathbf {D}}_1^{\mathrm{eff}}}{k_1+k_2} \end{aligned}$$
(38)

In a similar way, the same homogenized equation for \(c_{2}^{(0)}\) is obtained where both concentrations are linked by the equilibrium constraint \(k_1 c_1^{(0)} - k_2 c_2^{(0)}=0\) imposed everywhere in the fluid phase.

It is important to note that for this case, the effective diffusion tensor \({\mathbf {D}}^{\mathrm{eff}}_{\mathrm{H}}\) depends on the reaction rates when the microscopic diffusion coefficients are different and the two species share the same macroscopic mass conservation equation constrained by the chemical equilibrium condition (30).

The macroscopic equations in dimensional space according to the result 3 are given as follows.

Result 4

For diffusion and reaction of the same order corresponding to \(\mathrm{Da} = {\mathcal {O}}(\varepsilon ^{0})\), the concentrations at the leading order are solution of the homogenized coupled diffusion-reaction problem:

$$\begin{aligned} \begin{array}{ccc} \left\{ \begin{array}{rcl} \displaystyle \frac{\partial c_{i}^{*(0)}}{\partial t^*} - {{\varvec{\nabla }}}_{x^{*}}\cdot \left( {\mathbf {D}}_{\mathrm{H}}^{*\mathrm eff} \cdot {{\varvec{\nabla }}}_{x^{*}}c_i^{*(0)} \right) &{}=&{} 0 \quad \mathrm{with} \; i=1\; \text {or} \; 2\\ &{} &{} \\ k_1^*c_1^{*(0)} - k_2^*c_2^{*(0)}&{}=&{}0 \end{array} \right. \end{array} \end{aligned}$$
(39)

The homogenized tensor is given by

$$\begin{aligned} {\mathbf {D}}_{\mathrm{H}}^{*\mathrm eff} = \displaystyle \frac{k_1^*{\mathbf {D}}_2^{*\mathrm eff} + k_2^*{\mathbf {D}}_1^{*\mathrm eff}}{k_1^*+k_2^*} \end{aligned}$$
(40)

where the macroscopic diffusion tensors \({\mathbf {D}}_1^{*\mathrm eff}\) and \({\mathbf {D}}_2^{*\mathrm eff}\) are defined by relation (20).

6 Predominant Reaction

Let consider the last case \(\mathrm{Da}^{L} = {\mathcal {O}}(\varepsilon ^{-1})\) corresponding to a reaction rate much faster than the diffusion mechanism. An important question is to verify if the classical homogenization approach is able to take into account large Damköhler number and to capture the information corresponding to very short times of reaction (in comparison with the characteristic diffusion time).

For \(\text {Da} = {\mathcal {O}}(\varepsilon ^{-1})\), the \(\varepsilon\)-microscopic model to homogenize is written as

$$\begin{aligned} \left\{ \begin{array}{rcll} \displaystyle \frac{\partial c_1^{(\varepsilon )}}{\partial t} - {{\varvec{\nabla }}}\cdot ({\mathcal {D}}_1{{\varvec{\nabla }}}c_1^{(\varepsilon )}) &{}=&{} 0 &{} \text{ in } Y_{f} \\ \displaystyle \frac{\partial c_2^{(\varepsilon )}}{\partial t} - {{\varvec{\nabla }}}\cdot ({\mathcal {D}}_2{{\varvec{\nabla }}}c_2^{(\varepsilon )}) &{}=&{} 0 &{} \text{ in } Y_{f} \\ - {\mathcal {D}}_1{{\varvec{\nabla }}}c_1^{(\varepsilon )} \cdot {\mathbf {n}}_{fs} &{}=&{} \varepsilon ^{-1} (k_1 c_1^{(\varepsilon )} - k_2 c_2^{(\varepsilon )}) &{} \text{ at } \partial Y_{fs} \\ -{\mathcal {D}}_2{{\varvec{\nabla }}}c_2^{(\varepsilon )} \cdot {\mathbf {n}}_{fs} &{}=&{} \varepsilon ^{-1} (k_2 c_2^{(\varepsilon )} - k_1 c_1^{(\varepsilon )}) &{} \text{ at } \partial Y_{fs} \end{array} \right. \end{aligned}$$
(41)

leading to the following result.

Result 5

For a predominant reaction corresponding to \(\mathrm{Da}^{L} = {\mathcal {O}}(\varepsilon ^{-1})\), we obtain the same result as the previous case when the diffusion and the reaction are of the same order (see result 3 for details).

Proof

the proof of this result is detailed as follows. \(\square\)

\(\bullet\) Slow Variables

At order \({\mathcal {O}}(\varepsilon ^{-2})\) in the volume and \({\mathcal {O}}(\varepsilon ^{-1})\) at the interface, we have from (41):

$$\begin{aligned} \left\{ \begin{array}{rcll} {{\varvec{\nabla }}}_y \cdot ({\mathcal {D}}_1{{\varvec{\nabla }}}_y c_1^{(0)}) &{}=&{} 0 &{} \text{ in } Y_{f} \\ {{\varvec{\nabla }}}_y \cdot ({\mathcal {D}}_2{{\varvec{\nabla }}}_y c_2^{(0)}) &{}=&{} 0 &{} \text{ in } Y_{f} \\ -{\mathcal {D}}_1 {{\varvec{\nabla }}}_y c_1^{(0)} \cdot {\mathbf {n}}_{fs} &{}=&{} k_1 c_1^{(0)}-k_2 c_2^{(0)} &{} \text{ at } \partial Y_{fs} \\ -{\mathcal {D}}_2 {{\varvec{\nabla }}}_y c_2^{(0)} \cdot {\mathbf {n}}_{fs} &{}=&{} k_2 c_2^{(0)}-k_1 c_1^{(0)} &{} \text{ at } \partial Y_{fs} \end{array} \right. \end{aligned}$$
(42)

The uniqueness of the solution of equation (42) is now proved in a simple and formal manner, without considering the coercivity of the associated bilinear form in the relevant functional spacesFootnote 3. To do that, let define the auxiliary function \(C=k_1 c_1^{(0)} - k_2 c_2^{(0)}\). It is straightforward to show that C is solution of a Laplace problem with homogeneous Fourier boundary condition

$$\begin{aligned} \left\{ \begin{array}{rcll} \Delta _y C = 0 &{} \text {in } Y_{f}\\ {{\varvec{\nabla }}}_{y}C \cdot {\mathbf {n}}_{fs} + \alpha C = 0 \ &{} \text {at } \partial Y_{fs} \end{array} \right. \end{aligned}$$
(43)

with \(\alpha =\displaystyle { \frac{k_{1} {\mathcal {D}}_{2} + k_{2} {\mathcal {D}}_{1} }{{\mathcal {D}}_1 {\mathcal {D}}_2 } }\). Since \(\alpha\) is positive, the unique solution isFootnote 4\(C=0\), i.e \(k_1 c_1^{(0)} = k_2 c_2^{(0)}\). Consequently, from (42), we have straightforwardly \(c_1^{(0)}=c_1^{(0)}({\mathbf {x}}, t)\) and \(c_2^{(0)}=c_2^{(0)}({\mathbf {x}},t)\).

\(\bullet\) Closure Problem

At order \({\mathcal {O}}(\varepsilon ^{-1})\) in the volume and \({\mathcal {O}}(\varepsilon ^{-0})\) at the interface, it yields

$$\begin{aligned} \left\{ \begin{array}{rcll} {{\varvec{\nabla }}}_y \cdot \left[ {\mathcal {D}}_1({{\varvec{\nabla }}}_x c_1^{(0)}+{{\varvec{\nabla }}}_y c_1^{(1)})\right] &{}=&{} 0 &{} \text{ in } Y_{f} \\ {{\varvec{\nabla }}}_y \cdot \left[ {\mathcal {D}}_2({{\varvec{\nabla }}}_x c_2^{(0)}+{{\varvec{\nabla }}}_y c_2^{(1)})\right] &{}=&{} 0 &{} \text{ in } Y_{f} \\ -{\mathcal {D}}_1({{\varvec{\nabla }}}_x c_1^{(0)}+{{\varvec{\nabla }}}_y c_1^{(1)}) \cdot {\mathbf {n}}_{fs} &{}=&{} k_1 c_1^{(1)} - k_2 c_2^{(1)} &{} \text{ at } \partial Y_{fs} \\ -{\mathcal {D}}_2({{\varvec{\nabla }}}_x c_2^{(0)}+{{\varvec{\nabla }}}_y c_2^{(1)}) \cdot {\mathbf {n}}_{fs} &{}=&{} k_2 c_2^{(1)} - k_1 c_1^{(1)} &{} \text{ at } \partial Y_{fs} \end{array} \right. \end{aligned}$$
(46)

By linearity, the solution is sought in the form

$$\begin{aligned} \begin{array}{rlll} c_1^{(1)} &{}=&{} {\varvec{\chi }}_1 \cdot {{\varvec{\nabla }}}_x c_1^{(0)} + \widehat{c_1}(t, {\mathbf {x}}) \\ c_2^{(1)} &{}=&{} {\varvec{\chi }}_2 \cdot {{\varvec{\nabla }}}_x c_2^{(0)} + \widehat{c_2}(t, {\mathbf {x}}) \end{array} \end{aligned}$$
(47)

where \(\widehat{c_1}(t, {\mathbf {x}})\) and \(\widehat{c_2}(t, {\mathbf {x}})\) are constant in the unit cell satisfying \(k_1 \widehat{c_1}(t, {\mathbf {x}}) -k_2 \widehat{c_2}(t, {\mathbf {x}})=0\).

Then replacing (47) in (46) leads to the following closure problem for the vectors \({\varvec{\chi }}_1\) and \({\varvec{\chi }}_2\):

$$\begin{aligned} \left\{ \begin{array}{rcll} {{\varvec{\nabla }}}_y \cdot \left[ {\mathcal {D}}_1\left( {\varvec{I}} + \left( {{\varvec{\nabla }}}_y{\varvec{\chi }}_1\right) ^{{\mathsf {T}}}\right) \right] &{}=&{} 0 &{} \text{ in } Y_{f} \\ {{\varvec{\nabla }}}_y \cdot \left[ {\mathcal {D}}_2\left( {\varvec{I}} + \left( {{\varvec{\nabla }}}_y{\varvec{\chi }}_2\right) ^{{\mathsf {T}}}\right) \right] &{}=&{} 0 &{} \text{ in } Y_{f} \\ -{\mathcal {D}}_1 ({\varvec{I}} +{{\varvec{\nabla }}}_y {\varvec{\chi }}_1) \cdot {\mathbf {n}}_{fs} &{}=&{} k_1 ({\varvec{\chi }}_1 - {\varvec{\chi }}_2) &{} \text{ at } \partial Y_{fs} \\ -{\mathcal {D}}_2 ({\varvec{I}} +{{\varvec{\nabla }}}_y {\varvec{\chi }}_2) \cdot {\mathbf {n}}_{fs} &{}=&{} k_2 ({\varvec{\chi }}_2 - {\varvec{\chi }}_1) &{} \text{ at } \partial Y_{fs} \end{array} \right. \end{aligned}$$
(48)

The problem (48) leads to \({\varvec{\chi }}_{1}={\varvec{\chi }}_{2}={\varvec{\chi }}\) with a proof similar to the uniqueness of equation (42). As a consequence, the closure problem in this case is similar to the previous case (result 3).

\(\bullet\) Macroscopic Mass Conservation Equations

At order \({\mathcal {O}}(\varepsilon ^{0})\), we have

$$\begin{aligned} \left\{ \begin{array}{llll} &{}&{}\displaystyle \frac{\partial c_1^{(0)}}{\partial t} - {{\varvec{\nabla }}}_x \cdot \left[ {\mathcal {D}}_1({{\varvec{\nabla }}}_x c_1^{(0)}+{{\varvec{\nabla }}}_y c_1^{(1)}) \right] - {{\varvec{\nabla }}}_y \cdot \left[ {\mathcal {D}}_1({{\varvec{\nabla }}}_x c_1^{(1)}+{{\varvec{\nabla }}}_y c_1^{(2)}) \right] = 0 &{} \text{ in } Y_{f} \\ &{} &{} &{} \\ &{}&{}\displaystyle \frac{\partial c_2^{(0)}}{\partial t} - {{\varvec{\nabla }}}_x \cdot \left[ {\mathcal {D}}_2({{\varvec{\nabla }}}_x c_2^{(0)}+{{\varvec{\nabla }}}_y c_2^{(1)}) \right] - {{\varvec{\nabla }}}_y \cdot \left[ {\mathcal {D}}_2({{\varvec{\nabla }}}_x c_2^{(1)}+{{\varvec{\nabla }}}_y c_2^{(2)}) \right] = 0 &{} \text{ in } Y_{f} \\ &{} &{} &{} \\ &{}&{}- {\mathcal {D}}_1({{\varvec{\nabla }}}_x c_1^{(1)}+{{\varvec{\nabla }}}_y c_1^{(2)}) \cdot {\mathbf {n}}_{fs} = k_1 c_1^{(2)} - k_2 c_2^{(2)} &{} \text{ at } \partial Y_{fs} \\ &{} &{} &{} \\ &{}&{}- {\mathcal {D}}_2({{\varvec{\nabla }}}_x c_2^{(1)}+{{\varvec{\nabla }}}_y c_2^{(2)}) \cdot {\mathbf {n}}_{fs} = k_2 c_2^{(2)} - k_1 c_1^{(2)} &{} \text{ at } \partial Y_{fs} \end{array} \right. \end{aligned}$$
(49)

Averaging (49a–b) over the fluid phase of the unit cell, considering the interface condition (49c–d) using the divergence theorem and the periodicity condition yield

$$\begin{aligned} \left\{ \begin{array}{llll} \displaystyle \frac{\partial c_1^{(0)}}{\partial t} - {{\varvec{\nabla }}}_x \cdot \left( {\mathbf {D}}_1^{\mathrm{eff}} \cdot {{\varvec{\nabla }}}_x c_1^{(0)} \right) +f({\mathbf {x}},t) &{}=&{} 0 \\ &{} &{} &{} \\ \displaystyle \frac{\partial c_2^{(0)}}{\partial t} - {{\varvec{\nabla }}}_x \cdot \left( {\mathbf {D}}_2^{\mathrm{eff}} \cdot {{\varvec{\nabla }}}_x c_2^{(0)} \right) -f({\mathbf {x}},t) &{}=&{} 0 \end{array} \right. \end{aligned}$$
(50)

where \(f({\mathbf {x}},t)\) is defined by

$$\begin{aligned} f({\mathbf {x}},t) = \displaystyle \frac{1}{\mid Y_f \mid } \int _{\partial Y_{fs}}\left( k_1 c_1^{(2)} - k_2 c_2^{(2)}\right) \mathrm{d}A \end{aligned}$$
(51)

and the effective diffusion tensors \({\mathbf {D}}_1^{\mathrm{eff}}\) and \({\mathbf {D}}_2^{\mathrm{eff}}\) defined by

$$\begin{aligned} \begin{array}{llll} {\mathbf {D}}_1^{\mathrm{eff}} &{}=&{} \left\langle {\mathcal {D}}_1\left( {\varvec{I}} + \left( {{\varvec{\nabla }}}_y {\varvec{\chi }}\right) ^{{\mathsf {T}}}\right) \right\rangle ^f \\ {\mathbf {D}}_2^{\mathrm{eff}} &{}=&{} \left\langle {\mathcal {D}}_2 \left( {\varvec{I}} + \left( {{\varvec{\nabla }}}_y {\varvec{\chi }}\right) ^{{\mathsf {T}}}\right) \right\rangle ^{f} \end{array} \end{aligned}$$
(52)

By using the constraint \(k_1 c_1^{(0)} - k_2 c_2^{(0)}=0\), \(f({\mathbf {x}}, t)\) can be determined from (50)

$$\begin{aligned} \begin{array}{llll} f({\mathbf {x}},t)= & {} {{\varvec{\nabla }}}_x \cdot \left[ \displaystyle \frac{({\mathbf {D}}_2^{\mathrm{eff}}-{\mathbf {D}}_1^{\mathrm{eff}})k_1}{k_1+k_2} \cdot {{\varvec{\nabla }}}_x c_1^{(0)} \right] \end{array} \end{aligned}$$
(53)

Finally, the macroscopic mass conservation equation for \(c_1^{(0)}\) and \(c_2^{(0)}\)reads as

$$\begin{aligned} \begin{array}{llll} \displaystyle \frac{\partial c_i^{(0)}}{\partial t} - {{\varvec{\nabla }}}_x \cdot \left[ \left( \displaystyle \frac{k_1 {\mathbf {D}}_2^{\mathrm{eff}} + k_2 {\mathbf {D}}_1^{\mathrm{eff}}}{k_1+k_2}\right) \cdot {{\varvec{\nabla }}}_x c_i^{(0)} \right]= & {} 0 \end{array} \end{aligned}$$
(54)

for \(i=1,2\) together with the constraint \(k_1 c_1^{(0)} - k_2 c_2^{(0)}=0\) imposed everywhere in the fluid phase. This result is similar to the previous case, even if the expression of the source term in (50) is different from the one involved in (34). Indeed, the former obtained for \(\text {Da}^{L} = {\mathcal {O}}(\varepsilon ^0)\) involves \(\widehat{c_1}(t, {\mathbf {x}})\) and \(\widehat{c_2}(t,{\mathbf {x}})\) at order \({\mathcal {O}}(\varepsilon )\) (Eq. 35), whereas the latter obtained for \(\text {Da}^{L} = {\mathcal {O}}(\varepsilon ^{-1})\) involves \({c_1^{(2)}}\) and \({c_2^{(2)}}\) at order \({\mathcal {O}}(\varepsilon ^{2})\) (Eq. 51) and therefore has “a less important contribution” in the macroscopic diffusion equations.

Back to the dimensional forms, we obtain the following result.

Result 6

For a predominant reaction corresponding to \(\mathrm{Da}^{L} = {\mathcal {O}}(\varepsilon ^{-1})\), the same result is obtained as the previous case when the diffusion and the reaction are of the same order (see result 4).

7 Analysis of the Results

7.1 Comparison of the Models

The classical homogenization method provides two different homogenized models according to different orders of magnitude of the Damköhler number. In the first one for small Damköhler number, namely dominant diffusion, the macroscopic model is a two-equation model. Each species diffuses with an effective diffusion tensor only depending on the microscopic diffusion coefficients and on the geometry. The coupling between both equations is due to the reaction with a same volume source term with an opposite sign. In the second model, for reaction and diffusion of the same order of magnitude or reaction predominant, an instantaneous chemical equilibrium is imposed everywhere in the fluid phase. Moreover, both species share the same mass conservation law with the same effective diffusion tensor depending on the reaction rates. The two models will be numerically analyzed in the next subsection.

7.2 Numerical Simulations

The aim of this section is to check the reliability of the macroscopic models obtained from the homogenization technique to predict the diffusion-reaction process at the macroscale for different values of the Damköhler number. To do this, the homogenized models HM I corresponding to Result 2 and HM II corresponding to Result 4 or 6 will be compared to the direct numerical simulations of the pore scale model (PSM) given by (1). It is important to underline that the problem (39a) corresponding to the model HM II satisfies the chemical equilibrium through the penalization condition (39b). As a consequence, the diffusion equation for one concentration can be solved and the other one deduced from the equilibrium condition. However, the initial conditions for \(c_1^{*(0)}\) and \(c_2^{*(0)}\) must satisfy the chemical equilibrium to avoid a boundary layer in time. Imposing such compatible initial condition will not allow comparing correctly HM II to PSM. Indeed, it is relevant to impose initially a nonequilibrium state and to analyze how each model tends towards equilibrium. Therefore, instead of solving (54), problem (50) will be solved, satisfying the chemical equilibrium condition (39b). In (50), the source term is considered as an unknown, so that we have three equations for the three unknowns \((c_1^{*(0)}, c_2^{*(0)}, f^*({\mathbf {x}},t) )\). By solving (50), the mass conservation is always satisfied, regardless of the initial condition, as the concentrations instantaneously readjust to reach equilibrium from \(t=0^+\). For the sake of simplicity, in the following the superscript \(^{(0)}\) will be omitted.

The direct numerical simulations are carried out in a 2D-porous medium of size \(L \times l\) which is made of an array of N unit cells of size l. Each unit cell is composed of a solid inclusion of radius R located at its center and a fluid phase (see Fig. 2). The macroscopic models are solved in a corresponding effective medium of the same size.

Fig. 2
figure 2

Porous medium composed of N unit cells used in the direct numerical simulation and its effective one. \(N=20\) is used for the numerical simulations

Full size image

Periodic boundary conditions for the concentrations are imposed at the bottom (\(y_2^{*}=0\)) and at the topFootnote 5 (\(y_2^{*}=l\)). In addition, homogeneous flux boundary conditions (zero flux) are imposed at \(x_1^{*}=0\) and at \(x_1^{*}=L\). The interface conditions (1c) and (1d) of the PSM are applied on the solid–fluid interface represented by the inclusion surfaces. The physical and numerical parameters are given in Table 1. For the initial conditions, between \(x_1^{*}=0\) and \(x_1^{*}=L/2\), \(c_1^{*}(t^{*}=0) = a_1^{*}\) and \(c_2^{*}(t^{*}=0) = b_1^{*}\) are imposed while between \(x_1^{*}=L/2\) and \(x_1^{*}=L\), \(c_1^{*}(t^{*}=0) = a_2^{*}\) and \(c_2^{*}(t^{*}=0) = b_2^{*}\). The values of \(a_1^{*}\), \(b_1^{*}\), \(a_2^{*}\) and \(b_2^{*}\) are chosen to be at nonequilibrium, i.e. \(k_1^{*} c_1^{*}-k_2^{*} c_2^{*}\ne 0\). It should be noted that it is more relevant for numerical simulations to define the Damköhler numbers \(\mathrm {Da}_1\) and \(\mathrm {Da}_2\) at the pore scaleFootnote 6, so that the results are independent of the number of elementary cells considered. The numerical simulations are performed with Comsol Multiphysics software (Fig. 3).

Table 1 Physical and numerical parameters used in simulations
Full size table

Figure 4 displays the concentration fields obtained by HM I, II and PSM for different Damköhler number values and at several times. The concentrations \(c_1^{*}\) and \(c_2^{*}\) are normalized according to the maximum initial values \(\max \left( a^{*}_1, a^{*}_2 \right)\) and \(\max \left( b^{*}_1, b^{*}_2 \right)\), respectively. The results obtained by the PSM and the HM I are in very good agreement for low values of Damköhler number as the concentrations reach a constant value corresponding to the equilibrium state. For higher values of Damköhler number, the difference between the two models is observed mainly at short times. The HM I is only accurate for small values of Damköhler number. On the other hand, the HM II fails to predict the macroscopic behavior of the reaction-diffusion problem when the initial conditions do not satisfy the chemical equilibrium condition. This condition is strictly imposed in the HM II at every time and cannot be satisfied at \(t=0\).

Fig. 3
figure 3

Concentration \(c_2^{*}\) for \(\mathrm Da_1=10\). Comparison with PSM

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To better analyze the evolution of the concentrations, in Fig. 4\(c_1^{*}\) and \(c_2^{*}\) obtained from the PSM, the HM I and II are plotted at point \((x_{1}^{*} = 15 l \, , y_{2}^{*}= l/2)\) with respect to time. For small values of the Damköhler number (\(\mathrm{Da}_{1} \leqslant 1\)), the results obtained from HM I and PSM are similar, which validates the HM I in this range of Damköhler number. For \(\mathrm{Da}_{1}\ge 1\), the HM I overestimates the reaction rate leading to a faster establishment of the chemical equilibrium before reaching the steady state for \(t^{*}\geqslant 10^{-2}\). The results show also that the HM II fails to predict the evolution of the concentrations for short times, all concentration profiles being identical regardless of the Damköhler number \(\mathrm{Da}_{1}\). Nevertheless, the HM II is capable of predicting correctly the values of the concentrations of the two plateaux. The first one corresponds to the chemical equilibrium added to the mass conservation for the concentrations, whereas the second one corresponds to the steady state for long times.

Fig. 4
figure 4

Concentration vs time at point \((x_{1}^{*} = 15 l \, , y_{2}^{*} = l/2)\) and for different values of \(\mathrm{Da}\)

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In Fig. 5, the evolution of \(\frac{k_{1}^{*}c_{1}^{*}-k_{2}^{*}c_{2}^{*}}{k_{1}^{*}+k_{2}^{*}}\) with respect to time at point \((x_{1}^{*} = 15 l \, , y_{2}^{*}= l/2)\) for several values of \(\mathrm{Da}_{1}\) is displayed. Obviously, the HM II satisfies the imposed equilibrium at all time, whereas for the PSM and the HM I, the equilibrium is reached faster for higher \(\mathrm Da_1\). The difference between the results obtained by the PSM and the HM I becomes significant at high values of \(\mathrm{Da}_{1}\).

Fig. 5
figure 5

Evolution of \(\frac{k_{1}^{*}c_{1}^{*}-k_{2}^{*}c_{2}^{*}}{k_{1}^{*}+k_{2}^{*}}\) vs time for different values of \({\mathrm{Da}}_1\)

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Finally, to quantify the difference between the HM I and II and the PSM, the relative errors are defined as

$$\begin{aligned} E_{A}^{I}= & {} \left| \frac{\left( \left\langle c_{1}^{*}\right\rangle _{y_{2}^{*}}-c_{1}^{*\text {HM I}}\right) }{\left\langle c_{1}^{*}\right\rangle _{y_{2}^{*}}}\right| ,\qquad E_{A}^{II}=\left| \frac{\left( \left\langle c_{1}^{*}\right\rangle _{y_{2}^{*}}-c_{1}^{*\text {HM II}}\right) }{\left\langle c_{1}^{*}\right\rangle _{y_{2}^{*}}}\right| \end{aligned}$$
(55)
$$\begin{aligned} E_{B}^{I}= & {} \left| \frac{\left( \left\langle c_{2}^{*}\right\rangle _{y_{2}^{*}}-c_{2}^{*\text {HM I}}\right) }{\left\langle c_{2}^{*}\right\rangle _{y_{2}^{*}}}\right| ,\qquad E_{B}^{II}=\left| \frac{\left( \left\langle c_{2}^{*}\right\rangle _{y_{2}^{*}}-c_{2}^{*\text {HM II}}\right) }{\left\langle c_{2}^{*}\right\rangle _{y_{2}^{*}}}\right| \end{aligned}$$
(56)

where \(E_{A}^{I}\), \(E_{B}^{I}\), \(E_{A}^{II}\) and \(E_{B}^{II}\) refer to the relative errors of \(c_1^{*}\) and \(c_2^{*}\) obtained by the model HM I and II, respectively, according to the PSM, \(\left\langle c_{i}^{*} \right\rangle _{y_{2}^{*}}\) being the concentration averages over \(y_{2}^{*}\) given byFootnote 7

$$\begin{aligned} \left\langle c^{*}_i\right\rangle _{y^{*}_2}=\frac{1}{l}\int _{0}^{l} c^{*}_i dy^{*}_2 ,\quad i=1, 2 \end{aligned}$$
(57)

Figure 6 shows the evolution of the relative errors with respect to time for different values of \(\mathrm{Da}_{1}\) at the point \((x_{1}^{*} = 15 l \, , y_{2}^{*}= l/2)\). The relative error of the HM I increases with \(\text{Da}_{1}\) and is significant for short times. From \(\mathrm{Da}_{1}>1\), the HM I is not accurate to predict the coupled diffusion-reaction transport at the macroscale at short times.

Fig. 6
figure 6

Relative errors between the PSM and the HM I and II vs time evaluated at the point \((x_{1}^{*} = 15 l \, , y_{2}^{*}= l/2)\)

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The relative error of HM II is most important for short times. Higher the Damköhler number \(\text {Da}_{1}\) is, faster the error \(E^{II}_{A(B)}\) decreases with time. Indeed, the HM II reaches the chemical equilibrium instantaneously, whereas the PSM predicts the equilibrium state depending on the Damköhler number. Therefore, the gap between the PSM and the HM II progressively decreases towards zero when the PSM reaches the chemical equilibrium (see Fig. 5).

8 Conclusions

The classical homogenization procedure has been used to upscale the diffusion-heterogeneous reaction problem in a porous medium for different Damköhler numbers. The results accuracy has been numerically checked, leading to the following remarks:

  • For small order of magnitude of the Damköhler number (\(\mathrm{Da}^{L} = {\mathcal {O}}(\varepsilon )\)), the macroscopic model obtained by the homogenization technique is robust capable of describing accurately the coupled physics at the macroscale. In this case, the effective diffusion tensors depend only on the geometry and on the microscopic diffusion coefficients.

  • For higher Damköhler number (\(\mathrm{Da}^{L} \ge {\mathcal {O}}(\varepsilon ^0)\)), the macroscopic model predicts an instantaneous equilibrium everywhere in the fluid phase and the effective tensor depends on the reaction rates. Both species share the same macroscopic mass conservation equation at the macroscale. Numerical simulations have shown a considerable error of the homogenized models compared to direct numerical simulations of the pore-scale model, in particular for short times.

    This underlines the limitation of such classical homogenization approach that cannot take into account characteristic times of (very) different orders of magnitude involved in the physics at the local (micro) scale, which is the case here for high values of the Damköhler number, where the characteristic time of reaction is much faster than the diffusion one. More advanced study should be made to overcome this issue.