# OGS Conventions

## Units

In general OpenGeoSys is not using any specified intrinsic units, i.e., OGS assumes that a self-consistent set of units ( SI-units for example) is used for all quantities. However, there are some exceptions to that rule concerning empirical laws. For instance there are some expressions for the vapor diffusion and latent heat implemented, which assume that temperature is given in Kelvin. Therefore, we recommend just using SI units everywhere.

For more information on how to come up with a self-consistent unit scheme see this PDF.

### Dimensional analysis of source and boundary terms based on their primary variables

Most processes in OpenGeoSys are derived and implemented with a 3D spatial setting in mind. I.e., the conceptual model of the processes exists in a three-dimensional world and boundaries are two-dimensional. The canonical (a) Neumann boundary conditions and (b) volumetric source/sink terms (STs) for those processes are to (a) assign a flux (aka as flux density, i.e., a total flux $Q$ of some quantity (like volume of water per time) per a 2D pre-defined area) to parts of the boundary of the domain or to (b) add a quantity rate density (i.e., a total flux of some quantity $Q$ per 3D volume) to the simulation domain, respectively. Here, $Q$ in our formulation is an already integrated flux. Meaning it is typically a different quantity than the primary unknown of the process under consideration. It may be e.g. a secondary variable being a function of the primary unknown. This is for instance the case, if we consider prescribed fluxes of water in a porous medium on a Neumann boundary. Here, $Q$ would be a prescribed Darcy velocity integrated over some area being a function of the pressure gradient. For energy balances in OpenGeoSys the primary unknown is usually the temperature $T$, while corresponding fluxes $Q$ are usually expressed in terms of energy.

Note: The connection between the primary unknown and these canonical BCs and STs is made via the implemented partial differential equation (PDE). If a PDE is implemented slightly differently (e.g. PDE (B) is PDE (A) multiplied by a factor of density), the corresponding quantities in the BCs and STs generally also change (by a factor of density in this example).

Lower-dimensional realizations of the mentioned processes are essentially 2D slices through a 3D world. Hence, the parameterization does not change. Despite solving a problem whose solution varies only in one or two dimensions, you still have to apply Neumann BCs and volumetric STs as integrated quantity per time per 2D area and quantity per time per 3D volume, respectively. Sometimes, these lower dimensional realizations are referred to as cross-sectional models.

In contrast, for real two-dimensional and one-dimensional problems the fluxes of Neumann boundary conditions and volumetric source terms can be seen as normalized by their one- and two-dimensional boundaries, e.g., quantity per time per lengthd, where d stands for dimensionality, in the case of volumetric STs.

Attention: The specific units given below apply to 3D and mentioned cross-sectional models. That applies to all of OpenGeoSys’s processes; currently there are no processes in OpenGeoSys that are manifest 1D or 2D.

Units:

Quantity name Dimension symbol SI base units
time $\mathrm{T}$ $\mathrm{s}$
length $\mathrm{L}$ $\mathrm{m}$
mass $\mathrm{M}$ $\mathrm{kg}$
temperature $\mathrm{\Theta}$ $\mathrm{K}$

Temperature:

• Dirichlet BC: Boundaries held at a fixed temperature (Units: $\mathrm{\Theta}$, SI: $\mathrm{K}$).
• Neumann BC: Used to impose a heat flux through a boundary domain (Units: $\mathrm{M \cdot T^{-3}}$, SI: $\mathrm{W}\cdot \mathrm{m}^{-2}$).
• Volumetric ST: Representing a volumetric heat source in the model domain (Units: $\mathrm{M \cdot L^{-1} \cdot T^{-3}}$, SI: $\mathrm{W} \cdot \mathrm{m}^{-3}$).
• Nodal ST: Representing a heat source in the model domain (Units: $\mathrm{M \cdot L^{d-1} \cdot T^{-3}}$, SI: $\mathrm{W} \cdot \mathrm{m}^{d-3}$).
• Line ST: Representing a heat source on a line shaped subdomain (Units: $\mathrm{M} \cdot \mathrm{L}^{d-2} \cdot \mathrm{T}^{-3}$, SI: $\mathrm{W} \cdot \mathrm{m}^{d-4}$).

Displacement:

• Dirichlet BC: Boundaries held at a fixed displacement (Units: $\mathrm{L}$, SI: $\mathrm{m}$).
• Neumann BC: Used to apply an external traction at a boundary (Units: $\mathrm{M \cdot L^{-1} \cdot T^{-2}}$, SI: $\mathrm{N} \cdot \mathrm{m}^{-2}$).
• Volumetric ST: A body force density acting in the model domain. One example is gravity, but gravity can be modeled by other means in OpenGeoSys (Units: $\mathrm{M \cdot L^{-2} \cdot T^{-2}}$, SI: $\mathrm{N} \cdot \mathrm{m}^{-3}$).

Pressure:

• Dirichlet BC: Boundaries held at a fixed pressure (Units: $\mathrm{M \cdot L^{-1} \cdot T^{-2}}$, SI: $\mathrm{Pa}$).
• Neumann BC: Used to impose a mass flux through a domain boundary (Units: $\mathrm{M \cdot L^{-2} \cdot T^{-1}}$, SI: $\mathrm{kg} \cdot \mathrm{m}^{-2} \cdot \mathrm{s}^{-1}$).
• Volumetric ST: Representing a volumetric mass source in the model domain (Units: $\mathrm{ M \cdot L^{-3} \cdot T^{-1}}$, SI: $\mathrm{kg} \cdot \mathrm{m}^{-3} \cdot \mathrm{s}^{-1}$).
• Nodal ST: Representing a mass production rate in the model domain (Units: $\mathrm{M} \cdot \mathrm{L}^{d-3} \cdot \mathrm{T}^{-1}$, SI: $\mathrm{kg} \cdot \mathrm{m}^{d-3} \cdot \mathrm{s}^{-1}$).
• Line ST: Representing a mass source on a line shaped subdomain (Units: $\mathrm{M} \cdot \mathrm{L}^{d-4} \cdot \mathrm{T}^{-1}$, SI: $\mathrm{kg} \cdot \mathrm{m}^{d-4} \cdot \mathrm{s}^{-1}$).

Pressure (Liquid Flow Process):

The liquid flow process uses a volume-based PDE, not a mass-based one. Therefore, the list above does not apply to the liquid flow process, but you have to divide all units of natural boundary conditions and source terms by density, leading to the following list:

• Dirichlet BC: Boundaries held at a fixed pressure (Units: $\mathrm{M \cdot L^{-1} \cdot T^{-2}}$, SI: $\mathrm{Pa}$).
• Neumann BC: Used to impose a volume flux through a domain boundary (Units: $\mathrm{L} \cdot \mathrm{T}^{-1}$, SI: $\mathrm{m} \cdot \mathrm{s}^{-1} = \mathrm{m}^3 \cdot \mathrm{m}^{-2} \cdot \mathrm{s}^{-1}$).
• Volumetric ST: Representing a volumetric volume source in the model domain (Units: $\mathrm{T}^{-1}$, SI: $\mathrm{s}^{-1} = \mathrm{m}^{3} \cdot \mathrm{m}^{-3} \cdot \mathrm{s}^{-1}$).
• Nodal ST: Representing a volume production rate in the model domain (Units: $\mathrm{L}^{d} \cdot \mathrm{T}^{-1}$, SI: $\mathrm{m}^{d} \cdot \mathrm{s}^{-1}$).
• Line ST: Representing a volume source on a one-dimensional subdomain (Units: $\mathrm{L}^{d-1} \cdot \mathrm{T}^{-1}$, SI: $\mathrm{m}^{d-1} \cdot \mathrm{s}^{-1}$).

## Stress sign

OpenGeoSys uses the positive sign convention for tensile stresses, whereas compressional stresses carry a negative sign.

## Stress-strain assumption for 2D simulations

OpenGeoSys assumes plane strain conditions as default for 2D simulations.

## Order of process variables/global components

OpenGeoSys uses internally a vector of primary vector components (“global component vector”). Although, which set of components is used, varies from process to process, the order of primary variable components for most THM-based processes is: $T$, $p$, $u_{x}$, $u_{y}$, $u_{z}$. However, there are some exceptions and as there are also different primary variables for each process, you will find a list containing the process variables as they appear in the global component vector. This order is used, e.g., to display the per component convergence of the non-linear solver in the output.

## Symmetric tensors and Kelvin mapping

To map the elasticity/stiffness tensor OpenGeoSys internally uses a Kelvin mapping with an adapted component ordering for computational reasons

$[1](https://arxiv.org/abs/1605.09606)$

. For 2D, the Kelvin-Vector of the stress tensor looks like $\sigma=(\sigma_{xx},\sigma_{yy},\sigma_{zz},\sqrt{2}\sigma_{xy})$ whereas the 3D version reads as $\sigma=(\sigma_{xx},\sigma_{yy},\sigma_{zz},\sqrt{2}\sigma_{xy}, \sqrt{2}\sigma_{yz},\sqrt{2}\sigma_{xz})$.

For Kelvin mapping also see the conversion function documentation.

The input and output of symmetric tensors consists of the full (symmetric) tensor elements (without the factor $\sqrt{2}$), retaining the same order. I.e., Input and output components are $\sigma=(\sigma_{xx},\sigma_{yy},\sigma_{zz},\sigma_{xy})$ and $\sigma=(\sigma_{xx},\sigma_{yy},\sigma_{zz},\sigma_{xy},\sigma_{yz},\sigma_{xz})$, respectively.

## Staggered Scheme

A staggered scheme solves coupled problems by alternating on separate physical domains (e.g. thermal and mechanical) in contrast to monolithic schemes which solve all domains simultaneously (e.g., thermomechanical). Thus staggered schemes add another level of iterations, however this may pay off since the subproblems are smaller and they enable a finer tuning of the specific solvers.

### Fixed-stress Split for Hydro-mechanical Processes

For hydro-mechanical processes the fixed-stress split has been implemented, since it turned out advantageous [1]. For the sake of brevity, we do not describe the scheme itself, but intend to provide guidance for its stabilization parameter. On this parameter depends how many coupling iterations are needed and thus how long it takes to obtain a solution. The optimal value of this parameter is not a-priori known, only that it lies within a certain interval is known (see [2]):

$$\frac{1}{2}\frac{\alpha^2}{K_\mathrm{1D}} \le \beta_\mathrm{FS} \le \frac{\alpha^2}{K_\mathrm{ph}},$$

where $\alpha$ denotes Biot’s coefficient, and $K_\mathrm{1D}$ and $K_\mathrm{ph}$ are the bulk moduli described next. By $K_\mathrm{ph}$ we mean the bulk modulus adjusted to the spatial dimension, so in three dimensions it coincides with the drained bulk modulus, whereas in lower dimensions it becomes a constrained bulk modulus (2D plane strain, 1D uniaxial strain). Assuming isotropic, linear elasticity we have

$$\begin{eqnarray} K_\mathrm{3D} &=& \lambda + \frac{2}{3}\mu, \\ K_\mathrm{2D} &=& \lambda + \frac{2}{2}\mu, \\ K_\mathrm{1D} &=& \lambda + \frac{2}{1}\mu. \\ \end{eqnarray}$$

OGS sets the stabilization parameter, which corresponds to a coupling compressibility, via an optional tag fixed_stress_stabilization_parameter inside the tag coupling_scheme.

$$\beta_\mathrm{FS} = p_\mathrm{FS} \frac{\alpha^2}{K_\mathrm{3D}},$$

by default to $p_\mathrm{FS}=\frac{1}{2}$. For isotropic, linear elasticity we provide the interval [2] and the recommended value [3] in dependence on Poisson’s ratio $\nu$ (note $\frac{\lambda}{\mu}=\frac{2\nu}{1-2\nu}$).

2D 3D
$p_\mathrm{FS}^\mathrm{min}$ $\frac{1}{6}\frac{1+\nu}{1-\nu}$ same as 2D
$p_\mathrm{FS}^\mathrm{MW}$ $\frac{1+\nu}{3}$ $\frac{1}{2}$
$p_\mathrm{FS}^\mathrm{max}$ $\frac{2(1+\nu)}{3}$ $1$

For more information about the algorithms of the fixed stress splitting, please visit the page about a HM benchmark: Injection and Production in 1D Linear Poroelastic Medium with the Staggered Scheme.

## References

[1] Kim, J. and Tchelepi, H.A. and Juanes, R. (2009): Stability, Accuracy and Efficiency of Sequential Methods for Coupled Flow and Geomechanics. SPE International, vol. 16, p. 249--262, DOI:https://doi.org/10.2118/119084-PA https://onepetro.org/SJ/article-abstract/16/02/249/204235

[2] Storvik, E. and Both, J.W. and Kumar, K. and Nordbotten, J.M. and Radu, F.A. (2019): On the optimization of the fixed-stress splitting for Biots equations. International Journal for Numerical Methods in Engineering, vol. 120, p. 179--194, DOI:10.1002/nme.6130 https://onlinelibrary.wiley.com/doi/full/10.1002/nme.6130

[3] Mikelic, A. and Wheeler, M.F. (2013): Convergence of iterative coupling for coupled flow and geomechanics. Computational Geosciences, vol. 17, p. 455--461, DOI:https://doi.org/10.1007/s10596-012-9318-y https://link.springer.com/content/pdf/10.1007/s10596-012-9318-y.pdf

This article was written by Joerg Buchwald. If you are missing something or you find an error please let us know.
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