porousMicroTransport[^porousMicroTransport] is a set of additional solvers and related libraries for OpenFOAM developed for the purposes of simulating flow and transport in porous media, with an emphasis on paper-based microfluidics

Installation

porousMicroTransport requires OpenFOAM, as distributed by OpenCFD (openfoam.com). Compatible OpenFOAM versions are v2112, v2206, v2212, v2306, v2312, v2406, v2412 and v2506.

Versions produced by the OpenFOAM Foundation (openfoam.org) (e.g. OpenFOAM 11, OpenFOAM 12) are not compatible. macOS users may want to consider OpenFOAM.app.

With styro

porousMicroTransport can be installed with the styro package manager:

sh styro install porousmicrotransport

Install from source

Optional: reagency

porousMicroTransport can be used with the optional reagency library for reaction modeling. To install porousMicroTransport with reagency, clone the reagency repository alongside porousMicroTransport, and install the former first:

sh git clone https://github.com/gerlero/reagency.git reagency/Allwmake -j

Download

Download the source code of porousMicroTransport, or clone this repository with Git:

sh git clone https://github.com/gerlero/porousMicroTransport.git

Compile and install

To build and install porousMicroTransport, just invoke the top-level Allwmake script:

sh cd porousMicroTransport ./Allwmake -j

If necessary, activate/source the correct OpenFOAM environment before running Allwmake.

Test

Optionally, you can verify the installation of porousMicroTransport by running the included test suite (requires Python 3.7 or later):

sh tests/Alltest

Docker image

Alternatively, porousMicroTransport is also available in the form of Docker images. These images include porousMicroTransport precompiled and ready to use. Assuming Docker is installed, the following command will run the latest image and mount the current directory so that you can access the files inside:

sh docker run --rm -it -v $PWD:/root -w /root microfluidica/porousmicrotransport

Or, if you use OpenFOAM's openfoam-docker script (which takes care of making the working directory available inside the container):

sh openfoam-docker -image=microfluidica/porousmicrotransport

A slimmer image variant that does not include source code, development tools or tutorial cases is available as microfluidica/porousmicrotransport:slim.

Docker images can also be used with other compatible containerization software, such as Podman and Singularity/Apptainer.

Solvers

moistureDiffusivityFoam

(Unsaturated) capillarity-driven flow in a porous medium, governed by the moisture diffusivity equation[^Bear]:

$$\frac{\partial\theta}{\partial t} - \nabla\cdot\left[D\nabla\theta\right] = 0$$

where $\theta$ is the moisture content and $D$ is a saturation-dependent diffusivity as defined by an unsaturated flow model.

porousMicroTransportFoam

Transport by steady flow of any number of species in a porous medium, with optional reactions between the species. For each species (concentration $C$), the governing equation is:

$$\frac{\partial Rd \theta C}{\partial t} + \nabla\cdot\left[UC\right] - \nabla\cdot\left[\theta D{eff}\nabla C\right] = \theta F$$

where $F$ is a reaction term (see below), $R_d$ is defined as:

$$Rd = 1 + \frac{\rhos\left(1 - \varepsilon\textrm{tot}\right)Kd}{\theta}$$

and $D_{eff}$ is defined as:

$$D{eff} = \left(\frac{DM}{\tau} + \alphaT|V|\right)I + \left(\alphaL - \alpha_T\right)\frac{VV}{|V|}$$

with $I$ the identity tensor and $V$ the true velocity of the fluid ($=U/\theta$).

moistureDiffusivityTransportFoam

Capillary flow + reactive transport in a porous medium, coupling the moisture diffusivity equation for flow with the previous transport equation.

Case layout

The layout of porousMicroTransport cases follows many conventions of porousMultiphaseFoam, especially in field names and entries in the transportProperties dictionary. This allows for easy conversion of cases from porousMultiphaseFoam to porousMicroTransport (and to some extent, vice versa).

Common fields

These variable fields are defined in the time directories:

• theta: moisture content (scalar). Optional for porousMicroTransportFoam

• U: Darcy velocity (vector). Optional for flow solvers

Common porous medium properties

Defined as scalar fields in constant or as dictionary entries in transportProperties:

• eps or thetamax: effective porosity ($\varepsilon$)

• K: intrinsic permeability. Flow solvers only

• rs: particle density ($\rhos$). _Transport solvers only

• epsTotal: total porosity ($\varepsilon\textrm{tot}$). _Transport solvers only

• tau: diffusive tortuosity ($\tau$). Transport solvers only

• alphaT: transverse dispersion coefficient ($\alphaT$). _Transport solvers only

• alphaL: longitudinal dispersion coefficient ($\alphaL$). _Transport solvers only

Phase properties

Flow solvers only.

Set these in a phase.theta subdictionary in transportProperties:

• rho: density

• mu: dynamic viscosity

Moisture content options

Flow solvers only.

Defined as scalar fields in constant or as dictionary entries in transportProperties:

• thetamin: minimum (a.k.a. residual) moisture content

• thetamax: maximum moisture content (usually equal to the porosity)

Unsaturated flow models

Flow solvers only.

Supported models of unsaturated flow are:

• BrooksAndCorey: Brooks and Corey[^BrooksAndCorey] model

  • In coefficient dictionary BrooksAndCoreyCoeffs: pc0, alpha, n, l (optional)

• VanGenuchten: Van Genuchten[^VanGenuchten] model

  • In coefficient dictionary VanGenuchtenCoeffs: pc0, m or n, l (optional)

• LETxs: LETx + LETs model[^LET]

  • In coefficient dictionary LETCoeffs: pc0, Lw, Ew, Tw, Ls, Es, Ts

• LETd: LETd[^Validity] model

  • In coefficient dictionary LETCoeffs: pc0, L, E, T

To choose a model for your simulation, set the unsaturatedFlowModel entry in transportProperties. Then set the model-specific parameters in the corresponding coefficient subdictionary.

Special boundary conditions for flow

Flow solvers only.

Besides the standard OpenFOAM boundary conditions (e.g. zeroGradient, fixedValue), the solvers support these additional boundary conditions for theta:

• darcyGradPressure: follow the boundary condition set for velocity (same as darcyGradPressure in porousMultiphaseFoam).

• exhaustible: models an inlet reservoir with a fixed volume of fluid that is gradually depleted as fluid flows into the domain. A remaining entry is required (volume remaining in the reservoir).

Transported species

Transport solvers only.

A species list in transportProperties contains the names of all transported species.

Each species must also define its own scalar concentration field (named the same as the species).

For each species, the following entries can be set in transportProperties:

• Dm: molecular diffusivity ($D_M$)

• Kd: partitioning coefficient ($K_d$)

Reactions

Transport solvers only.

Reactions are defined in a reactions subdictionary in transportProperties. The reactions dictionary contains a list of subdictionaries, each of which defines a single reaction. A reaction can have an arbitrary name and should contain the following entries:

• reaction: reaction equation. E.g. "A^2 + B = 2C + D", where A, B, C and D are names of defined species

• kf: forward rate constant

• kr: optional reverse rate constant (for reversible reactions)

Automatic timestep control

To enable automatic timestep adjustment, set adjustTimeStep to yes in system/controlDict. Then, configure it as follows:

• For flow, set a tolerance value inside a Picard dictionary in system/fvSolution

• For transport, add a maxDeltaC and/or relMaxDeltaC entry in system/controlDict

Tutorials

Sample cases are available in the tutorials directory.

Related projects

• porousMultiphaseFoam[^porousMultiphaseFoam]: toolbox for OpenFOAM for modeling multiphase flow and transport. porousMicroTransport is mostly compatible with porousMultiphaseFoam in terms of case definitions, and can be installed alongside it.

• electroMicroTransport[^electroMicroTransport]: toolbox for OpenFOAM dedicated to electromigrative separations. It includes support for modeling separations in paper-based media, and can also be installed alongside porousMicroTransport.

[^porousMicroTransport]: Gerlero, G. S., Guerenstein, Z. I., Franck, N., Berli, C. L. A., & Kler, P. A. (2024). Comprehensive numerical prototyping of paper-based microfluidic devices using open-source tools. Talanta Open, 10, 100350. https://doi.org/10.1016/j.talo.2024.100350

[^Bear]: Bear, J., & Cheng, A. H. D. (2010). Modeling groundwater flow and contaminant transport (Vol. 23, p. 834). Dordrecht: Springer. https://doi.org/10.1007/978-1-4020-6682-5

[^porousMultiphaseFoam]: Horgue, P., Renard, F., Gerlero, G. S., Guibert, R., & Debenest, G. (2022). porousMultiphaseFoam v2107: An open-source tool for modeling saturated/unsaturated water flows and solute transfers at watershed scale. Computer Physics Communications, 273, 108278. https://doi.org/10.1016/j.cpc.2021.108278

[^electroMicroTransport]: Gerlero, G. S., Damián, S. M., & Kler, P. A. (2021). electroMicroTransport v2107: Open-source toolbox for paper-based electromigrative separations. Computer Physics Communications, 269, 108143. https://doi.org/10.1016/j.cpc.2021.108143

[^BrooksAndCorey]: Brooks, R., & Corey, T. (1964). Hydraulic properties of porous media. Hydrology Papers Colorado State University. https://mountainscholar.org/bitstream/handle/10217/61288/HydrologyPapers_n3.pdf

[^VanGenuchten]: Van Genuchten, M. T. (1980). A closed‐form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal, 44(5), 892-898. https://doi.org/10.2136/sssaj1980.03615995004400050002x

[^LET]: Lomeland, F. (2018). Overview of the LET family of versatile correlations for flow functions. In: Proceedings of the International Symposium of Core Analysts, SCA2018-056 http://www.jgmaas.com/SCA/2018/SCA2018-056.pdf

[^Validity]: Gerlero, G. S., Valdez, A. R., Urteaga, R., & Kler, P. A. (2022). Validity of capillary imbibition models in paper-based microfluidic applications. Transport in Porous Media, 141(2), 359-378. https://doi.org/10.1007/s11242-021-01724-w