Transport of Solute Concentration Fields
The command Transport Concentration Field employs a continuum-mechanical approach for solute transport, treating the concentration field as the main variable to be solved for. The model captures both advection, where the flowing solvent fluid carries the solute, and diffusion, driving the solute from regions of higher to lower concentration. The transport of a concentration profile, governed by advection, diffusion, or both mechanisms simultaneously, can be illustrated schematically as follows:
In this one-dimensional example, the fluid velocity was assumed constant.
The solution algorithm for solute transport is integrated into the LIR solver, and the flow field required for advection can be computed using LIR, EJ, or SimpleFFT solvers. The key features of the command Transport Concentration Field include:
- Stability (i.e., no artifacts or oscillations occur).
- Automatic time stepping (the user only specifies time intervals—so-called time batches).
- Minimal numerical dispersion.
- Bound preservation (implying non-negativity of concentrations).
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Note! It is possible that over- or undershoots (i.e. concentrations less than zero or greater than the maximum inflow concentration) occur on large, low-porosity structures. This is not caused by the transport solver, but due to the fact that the velocity field is not completely solenoidal (free of divergence). In the continuous case, holds. In the discrete case, this only holds up to the solver tolerance. For instance, when holds locally (i.e. for a LIR cell), more concentration flows into the cell than it flows out, which creates a local overshoot in . |
- Local mass conservation.
- Coverage of the full range of Péclet numbers, from pure diffusion (Pe = 0) to pure advection (Pe = ∞).
- Full compatibility with adaptive grids that stem from the flow computation with the LIR solver.
Governing Equations
The transport of solute concentration fields is calculated by solving the Advection–Diffusion equation:
The advection–diffusion equation consists of
- the evolutionary term ,
- the advection term , and
- the diffusion term .
The effective diffusivity is computed by:
where denotes the effective diffusivity in porous materials, obtained by multiplying the molecular diffusivity by the porosity and dividing by the tortuosity . For pure fluid voxels, the advection–diffusion equation simplifies, since both the porosity and the tortuosity are equal to one. The simplified equation reads:
This simplification is to be understood locally, i.e. voxel-wise (or, if the LIR solver is used, LIR-cell-wise, respectively). The advection-diffusion equation is solved in the union of fluid voxels and porous voxels (containing the active fluid).
Initial and Boundary Conditions
Since the problem is time-dependent, an initial condition for the concentration is required:
If advection is considered, the inflow concentration must be prescribed as the inflow boundary:
The inflow boundary is defined as all boundary points where the velocity vector points inward, i.e., . If advection but no diffusion is considered, the problem is closed and does not require further boundary conditions. If both advection and diffusion are considered, at outflow and no-flow boundaries, homogeneous Neumann conditions are applied:
where is the outward-pointing unit normal. This means that, at the outflow boundary, the diffusive flux is suppressed (homogeneous Neumann condition), while the advective flux remains free to leave the domain. In the case of pure diffusion, is assumed at all boundaries implying that no inflow or outflow boundaries exist.
