Fick's law of Diffusion

Fick's law describes the rate at which particles (such as molecules, atoms, or ions) diffuse through a medium. It helps us explain how substances spread and equalize concentrations in various contexts.

Fick's First Law

- The movement of particles from regions of high concentration to areas of low concentration (diffusive flux) is directly proportional to the particle’s concentration gradient.

- In simpler terms, a solute will move from an area of high concentration to a low concentration across a concentration gradient.

- Mathematically, Fick's first law can be expressed as

$$J=-D\frac{dC}{dx}$$

where:

-  (J) represents the diffusive flux (amount of substance passing through a unit area per unit time).

- (D) is the diffusion coefficient (specific to the substance and the medium).

- \(\frac{{dC}}{{dx}}\) is the concentration gradient (change in concentration with respect to distance).

Fick's Second Law

- This law predicts how the concentration gradient changes with time due to diffusion.

- It helps describe how the concentration profile evolves over time.

- The second law is derived from the first law and is identical to the diffusion equation,

$$\frac{{\partial C}}{{\partial t}} = D \nabla^2 C$$

Fickian Diffusion:

- When a diffusion process follows Fick’s laws, it is called normal or Fickian diffusion.

- In cases where diffusion deviates from Fick’s laws (such as diffusion through porous media or swelling penetrants), it is referred to as non-Fickian 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.