Steady-State Diffusion Process

Introduction

The Steady-State Diffusion Process solves the steady-state diffusion equation for pressure (or concentration) transport in porous media. This is a linear elliptic partial differential equation that models phenomena such as:

• Steady-state groundwater flow in porous media
• Heat conduction in solids
• Mass diffusion in continuous media
• Electrical potential distribution

Key Features

The Steady-State Diffusion process is linear, making it suitable for efficient solution using direct or iterative linear solvers. The diffusion coefficient can be specified as a tensor to model anisotropic diffusion behavior. The process supports calculation of surface fluxes through the calculatesurfaceflux feature, which can be used for mass balance verification. The surface flux is computed as specific_flux on the specified mesh. The process can output secondary variables such as darcy_velocity, the velocity field computed as $-\mathbf{k} \nabla p$.

Theoretical background

The Steady-State Diffusion Process solves the steady-state diffusion equation in strong form:

where:

• $p$ is the liquid phase pressure (primary variable)
• $\mathbf{k}$ is the diffusion coefficient tensor (material property)
• $Q$ is the source term (can be zero for no source)
• $\nabla$ is the gradient operator

Finite Element Discretization

The element stiffness matrix and load vector are computed as:

where $\mathbf{N}$ contains the shape functions and their gradients.

Definition in the project file

Steady-State Diffusion process has to be declared in the project file in the processes block. For example in following way:

<process>
    <name>SteadyStateDiffusion</name>
    <type>STEADY_STATE_DIFFUSION</type>
    <integration_order>2</integration_order>
    <process_variables>
        <process_variable>pressure</process_variable>
    </process_variables>
    <secondary_variables>
        <secondary_variable name="darcy_velocity"/>
    </secondary_variables>
    <calculatesurfaceflux> <!-- optional -->
        <mesh>boundary_mesh</mesh>
        <property_name>specific_flux</property_name>
    </calculatesurfaceflux>
</process>

For more detailed description of tags used in this snippet, please see Processes.

Process variables

The Steady-State Diffusion process requires single scalar process variable, e.g. pressure in the configuration above. For more details, see Process variables.

Medium properties

Required medium properties for each medium. See medium properties for more details on defining them.

Available benchmarks

To gain more insight into Steady-State Diffusion process, you can investigate Steady-State Diffusion benchmarks.