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Multibody approach for reactive transport modeling in discontinuous-heterogeneous porous media


In the context of long-term degradation of porous media, the coupling between fracture mechanics and reactive transport is investigated. A reactive transport model in a cracked discontinuous and heterogeneous porous medium is proposed. The species transport through and along the crack network are taken into account in a multibody approach. A dedicated geochemistry solver is developed and allows to take into account a significant number of chemical reactions. The model is validated via a benchmark for a porous medium without discontinuity. The applications deal with two kinds of chemical attacks in a pre-cracked concrete sample and highlight the impact of the discontinuities in the reactive transport kinetic and on the localization of the chemical degradation. The results bring out that a unidirectional descriptor, such as the depth of ingress rate, is not sufficient to describe the material degradation correctly.

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Appendix 1: Mass action law for solid reaction

The law of mass action is rewritten with logarithmic unknowns in order to ensure the positivity of all concentrations.

$$ \begin{array}{lc} \forall i \in [1,N^{sol}], \\ \displaystyle{\displaystyle{\xi_{i}^{sol}\left( \underline{C}^{aq}\right)} = \left( K_{i}^{sol}\right)^{-1} \prod\limits_{j=1}^{N^{aq,p}} \left( \gamma_{j} C^{aq,p}_{i} \right)^{S^{sol}_{i,j}} - 1 = 0 } \\ \displaystyle{\displaystyle{\xi_{i}^{sol}\left( \underline{C}^{aq}\right)} =} \\ \displaystyle{ \exp \left( -\ln\left( K_{i}^{sol}\right) + \sum\limits_{j=1}^{N^{aq,p}} S^{sol}_{i,j} \left( \ln(\gamma_{j}) + \ln\left( C^{aq,p}_{i}\right) \right) \right)} \end{array} $$

The bijectivity of the exponential function induces:

$$ \begin{array}{lc} \forall i \in [1,N^{sol}],\\ \displaystyle{\xi_{i}^{sol*}\left( \underline{C}^{aq}\right)} \overset{\text{def}}{=} \\ - \ln({K_{i}^{s}}) + {\sum}_{j=1}^{N^{aq,p}} S^{sol}_{i,j} \left( \ln(\gamma_{j}) + \ln\left( C^{aq,p}_{i}\right) \right) = 0 \end{array} $$

Appendix 2: Jacobian terms of the geochemical numerical method

The chemical model reads (see Section 3):

$$ \begin{array}{l} \text{find } \underline{C}^{aq,p} \in (\mathbb{R}_{+}^{\ast})^{N^{aq}}, \underline{C}^{ad,p} \in (\mathbb{R}_{+}^{\ast})^{N^{ad}} \text{ and } \\ \underline{C}^{sol} \in (\mathbb{R}_{+}^{\ast})^{N^{sol}} \text{ such that:} \\ \left\{\begin{array}{lc} \displaystyle{ \underline{{\varXi}}^{aq} \overset{\text{def}}{=} \underline{C}_{tot}^{aq} - \left[ \exp\left( \ln\left( \underline{C}^{aq,p}\right)\right) + \left( \underline {\underline{S}}^{aq}\right)^{T} \cdot \underline{C}^{aq,s} \right.}\\ \displaystyle{ \left.{\kern1.1cm} + \left( \underline {\underline{S}}^{ad/aq}\right)^{T} \cdot {C}^{ad,s} + \left( \underline {\underline{S}}^{sol}\right)^{T} \cdot\underline{C}^{sol} \right] = \underline{0} } \\ \displaystyle{ \underline{{\varXi}}^{ad} \overset{\text{def}}{=}} \\ \displaystyle{ \underline{C}_{tot}^{ad} - \left[ \exp(\ln(\underline{C}^{ad,p})) +\left( \underline {\underline{S}}^{ad/ad}\right)^{T} \cdot \underline{C}^{ad,s} \right] = \underline{0} } \\ \underline{{\varXi}}^{sol} \overset{\text{def}}{=} \underline{\xi}^{sol*} \left( \underline{C}^{aq,p} \right) = \underline{0} \\ \displaystyle{{\varXi}^{el} \overset{\text{def}}{=} \sum\limits_{i=1}^{N^{aq}_{p}+N^{aq}_{s}} z_{i} \exp\left( \ln\left( C_{i}^{aq}\right)\right) = 0 } \end{array}\right. \end{array} $$

The system is solved by a Newton-Raphson method. The Jacobian of this system is given by:

$$ \displaystyle{\underline {\underline{J}}\overset{\text{def}}{=}\begin{pmatrix} \frac{\partial {\underline{{\varXi}}^{aq}}}{\partial {\ln(\underline{C}^{aq,p})}} & \frac{\partial {\underline{{\varXi}}^{aq}}}{\partial {\ln(\underline{C}^{ad,p})}} & \frac{\partial {\underline{{\varXi}}^{aq}}}{\partial {\underline{C}^{sol}}} \\ \frac{\partial {\underline{{\varXi}}^{ad}}}{\partial {\ln(\underline{C}^{aq,p})}} & \frac{\partial {\underline{{\varXi}}^{ad}}}{\partial {\ln(\underline{C}^{ad,p})}} & \underline {\underline{0}} \\ -\frac{\partial {\underline{{\varXi}}^{sol}}}{\partial {\ln(\underline{C}^{aq,p})}} & \underline {\underline{0}} & \underline {\underline{0}} \end{pmatrix}} $$

where the Jacobian matrix terms are:

$$ \left\{ \begin{array}{lc} \displaystyle{\frac{\partial {{\varXi}_{i}^{aq}}}{\partial {\ln(C^{aq,p}_{j})}} =} \\ \displaystyle{ - \delta_{i,j} \exp(\ln(C_{i}^{aq,p})) - \sum\limits_{k=1}^{N_{s}^{aq}} S^{aq}_{j,k} \frac{\partial { C_{i}^{aq,s}}}{\partial {\ln(C^{aq,p}_{j})}}} \\ \displaystyle{ -\sum\limits_{k=1}^{N_{s}^{ad}} S^{ad}_{j,k} \frac{\partial {C_{k}^{ad,s}}}{\partial {\ln(C^{aq,p}_{j})}}} \\ \displaystyle{\frac{\partial {{\varXi}_{i}^{aq}}}{\partial {\ln(C^{ad,p}_{j})}} = - \sum\limits_{k=1}^{N_{s}^{ad}} S^{ad}_{j,k} \frac{\partial {C_{k}^{ad,s}}}{\partial {\ln(C^{ad,p}_{j})}}} \\ \displaystyle{\frac{\partial {\underline{{\varXi}}_{i}^{ad}}}{\partial {\ln(C_{i}^{aq,p})}} = -\sum\limits_{k=1}^{N_{s}^{ad}} S^{ad/ad}_{i,k} \frac{\partial {C_{k}^{ad,s}}}{\partial {\ln(C^{aq,p}_{j})}} } \\ \displaystyle{\frac{\partial {\underline{{\varXi}}_{i}^{ad}}}{\partial {\ln(C_{j}^{ad,p})}} =} \\ \displaystyle{ -\delta_{i,j} \exp(\ln(C_{i})) - \sum\limits_{k=1}^{N_{s}^{ad}} S^{ad/ad}_{i,k} \frac{\partial { C_{k}^{ad,s}}}{\partial {\ln(C^{ad,p}_{j})}} } \\ \displaystyle{\frac{\partial {\underline{{\varXi}}_{i}^{sol}}}{\partial {\ln(C^{aq,p}_{j})}} =} \\ \displaystyle{ - \sum\limits_{k=1}^{N^{ad}_{p}} S_{i,k}^{sol} \left( \frac{\partial {\ln(\gamma_{k})}}{\partial {\ln(C_{j}^{aq,p})}} + \delta_{k,j} \ln(C_{k}^{aq,p}) \right) } \\ \displaystyle{\frac{\partial {{\varXi}_{i}^{aq}}}{\partial {C^{sol}_{j}}} = - \sum\limits_{k=1}^{N_{}^{sol}} S^{sol}_{j,k} \delta_{k,j} } \end{array}\right. $$

with :

$$ \left\{ \begin{array}{lc} \displaystyle{\frac{\partial { C_{i}^{aq,s}}}{\partial {\ln(C^{aq,p}_{j})}} = \left( -\frac{\partial {\ln(\gamma_{i})}}{\partial {\ln(C^{aq,p}_{i})}} + \sum\limits_{k=1}^{N^{aq}_{p}} S_{i,k}^{aq} \left( \frac{\partial {\ln(\gamma_{k})}}{\partial {\ln(C_{j}^{aq,p})}} \right.\right.} \\ \displaystyle{\left.\left.{\kern2cm}+ \delta_{k,j} \ln(C_{k}^{aq,p}) \right) \right) C_{i}^{aq,s}} \\ \displaystyle{\frac{\partial { C_{i}^{ad,s}}}{\partial {\ln(C^{aq,p}_{j})}} = \left( \sum\limits_{k=1}^{N^{aq}_{p}} S_{i,k}^{ad/aq} \left( \frac{\partial {\ln(\gamma_{k})}}{\partial {\ln(C^{aq,p}_{j})}} \right.\right.} \\ \displaystyle{\left.\left.{\kern2. cm}+ \delta_{k,j} \ln\left( C^{aq,p}_{k}\right) \right) \right) C^{ad,s}_{i} } \\ \displaystyle{\frac{\partial { C^{ad,s}_{i}}}{\partial {\ln(C^{ad,p}_{j})}} = \left( \sum\limits_{k=1}^{N^{ad}_{p}} S_{i,k}^{ad/ad} \delta_{k,j} \ln(C^{ad,p}_{k}) \right) C^{ad,s}_{i}} \\ \frac{\partial {\ln(\gamma_{i})}}{\partial {\ln(C^{aq,p}_{j})}} = -A {z_{i}^{2}} \left( \frac{1}{2\sqrt{IS}\left( 1+\sqrt{IS}\right)^{2}} - b \right) \\ {\kern2cm} \times \delta_{i,j} {z_{j}^{2}} \exp(\ln(C_{j}^{aq,p})) \end{array}\right. $$

If the electroneutrality is imposed (26), the chemical system is overdetermined. As implemented in CHESS software [63], one mass conservation relations of the aqueous species is replaced by the electroneutrality equation. The associated Jacobian term is:

$$ \begin{array}{lc} \displaystyle{\frac{\partial {{\varXi}^{el}}}{\partial {\ln(C_{i}^{aq})}} = z_{i} \exp(\ln(C_{i}^{aq}))} & \forall i \in [1,N^{aq-1}] \end{array} $$

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Socié, A., Dubois, F., Monerie, Y. et al. Multibody approach for reactive transport modeling in discontinuous-heterogeneous porous media. Comput Geosci 25, 1473–1491 (2021).

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  • Reactive transport
  • Cracked porous media
  • Geochemistry
  • Zero-thickness interface
  • Multibody approach