We consider a porous medium with a pore space that is completely filled
by three different phases: two immiscible fluids (say water and oil)
and a solid phase. One fluid phase contains dissolved ions, which
can precipitate at the pore boundary to form the solid phase. The
reverse process of dissolution, is also possible. Consequently, the
solid phase changes in time; its variation is not known a-priori.
The second fluid contains no solute and has no interaction with the
solid phase. Starting from a standard sharp interface model for the
pore-scale dynamics we develop a diffuse interface approach that
accounts for the time-dependent spatial distribution of the three
species and the overall concentration of the solute. Basic analytical
results for this model are presented, including the well posedness
of the phase field component of the model. Next we apply matched
asymptotic techniques to show that the diffuse interface model converges
to the sharp interface one. Further, homogenization is applied to
derive a new two-scale model that is valid at the Darcy scale. This
leads to a parabolic reaction-diffusion system in a medium with variable,
concentration dependent porosity. The diffuse interface approach
allows describing the variation in the porosity as phase field type
equations at the pore-scale. The last part of the paper presents
an efficient numerical scheme to approximate the solution of the
two-scale model. This scheme has as starting point the algorithm
in M. Redeker and C. Eck. A fast and accurate adaptive solution
strategy for two-scale models with continuous inter-scale dependencies.
J. Comput. Phys., 240:268�283, 2013. After some test cases validating
the method, we finally present computations for several realistic
scenarios. The results demonstrate the interdependence of the change
of the pore structure due to precepitaion/dissolution and the evolution
of the Darcy scale concentration of the dissolved particles in the
one fluid.
%0 Journal Article
%1 redeker2016upscaling
%A Redeker, Magnus
%A Pop, Iuliu Sorin
%A Rohde, Christian
%D 2016
%J IMA J. Appl. Math.
%K from:sylviazur ians imported vorlaeufig
%P 898-939
%R https://doi.org/10.1093/imamat/hxw023
%T Upscaling of a Tri-Phase Phase-Field Model for Precipitation in Porous
Media
%U https://academic.oup.com/imamat/article-lookup/doi/10.1093/imamat/hxw023
%V 81(5)
%X We consider a porous medium with a pore space that is completely filled
by three different phases: two immiscible fluids (say water and oil)
and a solid phase. One fluid phase contains dissolved ions, which
can precipitate at the pore boundary to form the solid phase. The
reverse process of dissolution, is also possible. Consequently, the
solid phase changes in time; its variation is not known a-priori.
The second fluid contains no solute and has no interaction with the
solid phase. Starting from a standard sharp interface model for the
pore-scale dynamics we develop a diffuse interface approach that
accounts for the time-dependent spatial distribution of the three
species and the overall concentration of the solute. Basic analytical
results for this model are presented, including the well posedness
of the phase field component of the model. Next we apply matched
asymptotic techniques to show that the diffuse interface model converges
to the sharp interface one. Further, homogenization is applied to
derive a new two-scale model that is valid at the Darcy scale. This
leads to a parabolic reaction-diffusion system in a medium with variable,
concentration dependent porosity. The diffuse interface approach
allows describing the variation in the porosity as phase field type
equations at the pore-scale. The last part of the paper presents
an efficient numerical scheme to approximate the solution of the
two-scale model. This scheme has as starting point the algorithm
in M. Redeker and C. Eck. A fast and accurate adaptive solution
strategy for two-scale models with continuous inter-scale dependencies.
J. Comput. Phys., 240:268�283, 2013. After some test cases validating
the method, we finally present computations for several realistic
scenarios. The results demonstrate the interdependence of the change
of the pore structure due to precepitaion/dissolution and the evolution
of the Darcy scale concentration of the dissolved particles in the
one fluid.
@article{redeker2016upscaling,
abstract = {We consider a porous medium with a pore space that is completely filled
by three different phases: two immiscible fluids (say water and oil)
and a solid phase. One fluid phase contains dissolved ions, which
can precipitate at the pore boundary to form the solid phase. The
reverse process of dissolution, is also possible. Consequently, the
solid phase changes in time; its variation is not known a-priori.
The second fluid contains no solute and has no interaction with the
solid phase. Starting from a standard sharp interface model for the
pore-scale dynamics we develop a diffuse interface approach that
accounts for the time-dependent spatial distribution of the three
species and the overall concentration of the solute. Basic analytical
results for this model are presented, including the well posedness
of the phase field component of the model. Next we apply matched
asymptotic techniques to show that the diffuse interface model converges
to the sharp interface one. Further, homogenization is applied to
derive a new two-scale model that is valid at the Darcy scale. This
leads to a parabolic reaction-diffusion system in a medium with variable,
concentration dependent porosity. The diffuse interface approach
allows describing the variation in the porosity as phase field type
equations at the pore-scale. The last part of the paper presents
an efficient numerical scheme to approximate the solution of the
two-scale model. This scheme has as starting point the algorithm
in [M. Redeker and C. Eck. A fast and accurate adaptive solution
strategy for two-scale models with continuous inter-scale dependencies.
J. Comput. Phys., 240:268�283, 2013]. After some test cases validating
the method, we finally present computations for several realistic
scenarios. The results demonstrate the interdependence of the change
of the pore structure due to precepitaion/dissolution and the evolution
of the Darcy scale concentration of the dissolved particles in the
one fluid.},
added-at = {2019-11-11T13:34:59.000+0100},
author = {Redeker, Magnus and Pop, Iuliu Sorin and Rohde, Christian},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/25b7b93ab8d3a0c7552456d0c8100d322/mathematik},
doi = {https://doi.org/10.1093/imamat/hxw023},
interhash = {65b921ae8aa318bd2fcbdf3e7cf87f0d},
intrahash = {5b7b93ab8d3a0c7552456d0c8100d322},
journal = {IMA J. Appl. Math.},
keywords = {from:sylviazur ians imported vorlaeufig},
owner = {redeker},
pages = {898-939},
timestamp = {2019-12-18T14:37:55.000+0100},
title = {Upscaling of a Tri-Phase Phase-Field Model for Precipitation in Porous
Media},
url = {https://academic.oup.com/imamat/article-lookup/doi/10.1093/imamat/hxw023},
volume = {81(5)},
year = 2016
}