3D Convection-Diffusion Equation

3D Convection-Diffusion Equation#

This example solves the three-dimensional convection-diffusion equation using mimetic methods. The convection-diffusion equation is a parabolic partial differential equation that describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection.

Mathematical Model#

The convection-diffusion equation has the form:

\[\frac{\partial C}{\partial t} + \nabla \cdot (\mathbf{v} C) = \nabla \cdot (D \nabla C)\]

where:

  • \(C\) is the concentration (or density)

  • \(\mathbf{v}\) is the velocity field

  • \(D\) is the diffusion coefficient

This equation combines:

  • The diffusion term: \(\nabla \cdot (D \nabla C)\)

  • The convection term: \(\nabla \cdot (\mathbf{v} C)\)

Application Context#

This example simulates CO₂ transport in a geological formation with impermeable shale layers. The simulation represents CO₂ injection through a well, and its subsequent movement through porous media with varying permeability. This type of simulation is important for:

  • Carbon capture and storage (CCS) studies

  • Underground contaminant transport

  • Enhanced oil recovery analysis

Numerical Method#

The equation is solved using operator splitting with:

  1. A Forward-Time Central-Space (FTCS) scheme for the diffusion term

  2. An upwind scheme for the convection term

Mimetic operators are used for spatial discretization:

  • Divergence operator (\(D\))

  • Gradient operator (\(G\))

  • Interpolation operator (\(I\))

Time step constraints include:

  • von Neumann stability criterion for diffusion: \(\Delta t \leq \frac{\Delta x^2}{3D}\)

  • CFL condition for convection: \(\Delta t \leq \frac{\Delta x}{\max(|\mathbf{v}|)}\)

This example is implemented in:

Results#

The simulation shows how CO₂ spreads through the domain, with the shale layers acting as barriers to flow. The concentration profile evolves over time due to:

  1. Molecular diffusion (spreading in all directions)

  2. Advective transport (preferential movement in the direction of flow)

  3. Reduced transport through low-permeability layers

This type of simulation is valuable for understanding subsurface fluid dynamics and designing effective carbon storage strategies.

Contents