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Solute transport in heterogeneous reservoirs: Upscaling from the Darcy to the reservoir scale

Aguilar-Madera, Carlos G., Herrera-Hernández, Erik C., Espinosa-Paredes, Gilberto
Advances in water resources 2019 v.124 pp. 9-28
equations, geometry, mass transfer, models, porous media, solutes, water resources
In this work, we present a methodology for upscaling solute transport in heterogeneous reservoirs. The problem considers upscaling from the Darcy scale of a porous medium to the reservoir scale. We first define the mathematical problem that governs solute mass transport at the so-called Darcy scale, and then we apply the method of volume averaging. This method allows for upscaling the governing equations at the whole reservoir scale and, eventually, developing physical constraints, definitions of average variables and effective coefficients. One important characteristic of the volume averaging method is that the so-called closure problems for analytical or numerical computation of effective coefficients are identified. The closure problems consist of boundary-value problems, whose solution must be carried out in representative geometries (unit cells) of the reservoir. In this paper, we developed the theory for a two-facies reservoir, but the extension for more facies is straightforward. In this context, we present the two-equation model, and the numerical estimation of effective coefficients are presented and discussed for lenticular and layered arrangements of facies. In this manner, relevant coefficients are identified. The numerical solution of the upscaled model agrees well with direct numerical results when the governing equations are solved at the Darcy scale.