Fick's law of Diffusion

Hydrodynamic Dispersion, Random-walk models & Advection-Diffusion Equation

Hydrodynamic Dispersion:

Definition:

Hydrodynamic dispersion is the combined effect of molecular diffusion and mechanical dispersion in porous media, such as groundwater flow through soil or rock.

Components:

Advection: This represents the bulk movement of solutes due to the flow of water (pore water velocity, denoted as v).

Dispersion: Dispersion includes mechanical dispersion (due to variations in flow paths) and molecular diffusion (random movement of solute particles).

Advection-Diffusion Equation:

Purpose:

The advection-diffusion equation describes the transport of solutes (e.g., chemicals, heat, or contaminants) in a fluid medium.

Applicability:

The advection-diffusion equation is commonly used for modelling solute transport in porous media, including groundwater flow. It considers both advection (bulk flow) and diffusion (dispersion).

Components:

Advection: Represents the advective transport due to fluid flow.

Diffusion: Describes the dispersion caused by molecular diffusion.

Mathematical Representation:

$$\frac{\partial C}{\partial t}=-\:v\frac{\partial C}{\partial x}+D_{h}\frac{\partial^{*}C}{\partial x^{2}}$$

Random Walk Models:

Concept:

Random walk models simulate the movement of individual particles (e.g., radioactive nuclides) as they undergo random steps.

Advantages:

- Particle-level detail: Tracks individual trajectories.

- Captures local heterogeneity and tortuosity.

Disadvantages:

- Computationally intensive for large-scale systems.

- Assumes isotropic diffusion.

Application:

Useful for studying particle dispersion in complex geological formations.
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Why do we choose random walk models over hydrodynamic dispersion?

1. Particle-Level Detail:

- Hydrodynamic dispersion models treat solutes as continuous concentrations, averaging the entire porous medium.

- Contrastingly, random walk models track individual particles, capturing local heterogeneity and tortuosity.

- Understanding their behaviour at the particle level is crucial for radioactive nuclides.

2. Complexity and Scale:

- Hydrodynamic dispersion equations involve macroscopic parameters (e.g., dispersivity, flow velocity).

- Random walk simulations provide finer-scale insights but can be computationally intensive.

- Balancing both approaches is essential.

3. Isotropic Assumption:

- Hydrodynamic dispersion assumes isotropic diffusion (equal in all directions).

- Geological formations exhibit anisotropy due to varying grain sizes, fractures, and bedding planes.

- Random walk models can account for anisotropy more effectively.

4. Local Variability:

- Hydrodynamic dispersion coefficients are spatially averaged.

- Random walk models allow exploration of local variations in transport behaviour.

- Especially relevant for heterogeneous aquifers.

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Recommendation:

Consider integrating both frameworks:

- Use the advection-diffusion equation for bulk transport.

- Employ random walk simulations to capture particle-scale dynamics.