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\begin{document}

\title{A Generalized Mixed-Cell Method for Eulerian Network Transport Modeling}



\author[1]{Toko Kamtchueng}%
\author[2]{Adil Mohammed Sbai}%
\author[3]{Jean-Louis Rouet}%
\author[3]{Olivier Rozenbaum}%
\author[4]{Mohamed Azaroual}%
\affil[1]{BRGM}%
\affil[2]{BRGM (French Geological Qurvey)}%
\affil[3]{Université d'Orléans}%
\affil[4]{Bureau de Recherches Geolog. et Minieres}%


\vspace{-1em}



  
  \date{May 07, 2024}


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\begin{abstract}
Pore network models are efficient tools for upscaling flow and transport
properties in porous media. This work introduces a new formal
mathematical derivation of the discrete equations governing solute
transport in a pore network model. A double Laplace transform technique
is applied by enforcing mass flux continuity along with the interfaces
between pores and throats. A non-local semi-analytical formulation
results. Solutions are given as a sum of convolution products with
time-dependent and exponentially decaying local P\selectlanguage{ngerman}éclet-dependent
infinite series. Continuous concentration profiles along throats are
calculated analytically, a posteriori, from time-dependent numerically
simulated concentrations in pores. The upwind and central-difference
schemes of the widely used mixed-cell method are found to be equivalent
to the asymptotic form of this new formulation for the advective and
diffusive dominant regimes, respectively. Therefore, the validity range
of these static methods is established. The model was compared to the
delay differential equations approach, a newly derived analytical
solution, and mixed-cell methods on idealized one-dimensional networks
eliminating topological disorder. Concentrations in pores are best
reproduced when transport in the throats is not neglected unlike for the
mixed-cell method leading to early breakthrough and first-order moment.
An efficient numerical scheme truncating the encoded memory effects in
the convolution kernels is introduced. This paves the way for the model
application to realistic networks extracted from digital rock images. We
caution against using static formulations as the error can be very large
locally before attending a steady-state.%
\end{abstract}\selectlanguage{ngerman}%



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