Single-phase flow and tracer transport

In this example, single-phase flow and tracer transport through a domain with a heterogeneous permeability distribution is considered. A velocity distribution is obtained from the solution of a stationary single-phase problem, and subsequently, this velocity field is used for the simulation of the transport of tracer through the domain.

The main points illustrated in this example are

• setting up and solving a stationary single-phase flow problem
• setting up and solving a tracer transport problem
• solving two problems sequentially and realizing the data transfer
• using a simple method to generate a random permeability field

Table of contents. This description is structured as follows:

• Single-phase flow and tracer transport
  • Problem set-up
  • Model description
    • 1p Model
    • Tracer Model
    • Discretization
• Implementation
  • Part 1: Single-phase flow simulation setup
  • Part 2: Tracer transport simulation setup
  • Part 3: Main program flow

Problem set-up

A domain with an extent of 10 \, \mathrm{m} \times 10 \, \mathrm{m} is considered, in which a heterogeneous permeability distribution is generated randomly. The problem set-up is shown in the figure below. In the stationary single-phase simulation, a pressure difference between the bottom and the top boundaries is prescribed (see left figure), resulting in a non-uniform velocity distribution due to the heterogeneous medium. Neumann no-flow conditions are used on the lateral sides. On the basis of the resulting velocity field, the transport of an initial tracer concentration distribution through the domain is simulated (see right figure). Initially, non-zero tracer concentrations are prescribed on a small strip close to the bottom boundary.

Model description

As mentioned above, two models are solved sequentially in this example. A single-phase model (1p model) is used to solve for the stationary velocity distribution of a fluid phase in the domain. The tracer transport is solved with the tracer model, which solves an advection-diffusion equation for a tracer component, which is assumed not to affect the density and viscosity of the fluid phase.

1p Model

The single phase model uses Darcy's law as the equation for the momentum conservation:

\textbf v = - \frac{\textbf K}{\mu} \left(\textbf{grad}\, p - \varrho {\textbf g} \right),

with the darcy velocity \textbf v, the permeability \textbf K, the dynamic viscosity \mu, the pressure p, the density \varrho and the gravitational acceleration \textbf g.

Darcy's law is inserted into the mass balance equation:

\phi \frac{\partial \varrho}{\partial t} + \text{div} \left( \varrho \textbf v \right) = 0,

where \phi is the porosity. The primary variable used in this model is the pressure p.

Tracer Model

The tracer model solves the mass conservation equation of a tracer component \kappa, in which both advective and diffusive transport mechanisms are considered:

\phi \frac{ \partial \varrho X^\kappa}{\partial t} - \text{div} \left\lbrace \varrho X^\kappa {\textbf v} + \varrho D^\kappa_\text{pm} \textbf{grad} X^\kappa \right\rbrace = 0.

Here, \textbf v is a velocity field, which in this example is computed using the 1p model (see above). Moreover, X^\kappa is the tracer mass fraction and D^\kappa_\text{pm} is the effective diffusivity. In this example, the effective diffusivity is a function of the diffusion coefficient of the tracer component D^\kappa and the porosity and tortuosity \tau of the porous medium (see dumux/material/fluidmatrixinteractions/diffusivityconstanttortuosity.hh):

D^\kappa_\text{pm}= \phi \tau D^\kappa.

The primary variable used in this model is the tracer mass fraction X^\kappa.

Discretization

In this example, all equations are discretized using cell-centered finite volumes with two-point flux approximation as spatial discretization scheme. For details on the discretization schemes available in DuMuX, have a look at the handbook. We use the implicit Euler method as time discretization scheme for the tracer component balance equation solved in the tracer model.

Implementation

In the following, we take a closer look at the source files for this example:

└── 1ptracer/
    ├── CMakeLists.txt          -> build system file
    ├── main.cc                 -> main program flow
    ├── params.input            -> runtime parameters
    ├── properties._1p.hh       -> compile time settings for the single-phase flow simulation
    ├── problem_1p.hh           -> boundary & initial conditions for the single-phase flow simulation
    ├── spatialparams_1p.hh     -> parameter distributions for the single-phase flow simulation
    ├── properties_tracer.hh    -> compile time settings for the tracer transport simulation
    ├── problem_tracer.hh       -> boundary & initial conditions for the tracer transport simulation
    └── spatialparams_tracer.hh -> parameter distributions for the tracer transport simulation

In order to define a simulation setup in DuMuX, you need to implement compile-time settings, where you specify the classes and compile-time options that DuMuX should use for the simulation. Moreover, a Problem class needs to be implemented, in which the initial and boundary conditions are specified. Finally, spatially-distributed values for the parameters required by the used model are implemented in a SpatialParams class.

In the documentations behind the links provided in the following, you will find how the above- mentioned settings and classes are realized in this example. Finally, it is discussed how the two simulations are solved sequentially in the main file (main.cc).

Part 1: Single-phase flow simulation setup

Part 2: Tracer transport simulation setup

Part 3: Main program flow