6.2 Advection | CE 574: Reactive Transport in Porous Media

Advection is the transport process where solutes flow with the bulk fluid phase. This is like you let go of yourself when you swim so you have the same velocity of the flowing fluid. The advective flux, $J_{a d v} (mol / m^{2} / s)$ of the solute, can be expressed as

J_{a d v} = ϕ v C

(1)

where $ϕ$ is the porosity of porous media; v is the linear fluid velocity in poroud media (m/s); and C is the solute concentration (mol/m³). Flow through a porous medium is described with Darcy’s Law:

u = ϕ v = - K \cdot \nabla h

(2)

where $u$ is the Darcy flux $(m_{fluid}^{3} / m_{medium}^{2} / s)$ that is proportional to the gradient in the hydraulic head $\nabla h (m)$ ; K is the hydraulic conductivity (m/s); One can also write Darcy’s Law in terms of hydraulic head by defining the hydraulic head as

h = z + \frac{P}{ρ g}

(3)

where z is the depth (m), P is the fluid pressure (Pa), $ρ$ is the fluid density (kg/m³), and g is the acceleration of gravity (9.8 N/m²). The hydraulic conductivity (m/s) can be expressed as

K = \frac{κ ρ g}{μ}

(4)

where $κ$ is the permeability of the porous media (m²) and is independent of fluid property, $μ$ is the fluid hydraulic viscosity (Pa·s). Representative values of hydraulic conductivity and permeability are listed in Table 1 for various subsurface materials.

Table 1. Values of hydraulic conductivity and permeability for different subsurface materials. Modified from [*Bear*, 2013]
K (m/s)	10⁰	10^-1	10^-2	10^-3	10^-4	10^-5	10^-6	10^-7	10^-8	10^-9	10^-10	10^-11	10^-12
κ (m²)	10^-7	10^-8	10^-9	10^-10	10^-11	10^-12	10^-13	10^-14	10^-15	10^-16	10^-17	10^-18	10^-19
Unconsolidated Sand & Gravel	Clean Gravel		Clean Sand or Sand & Gravel			Very Fine Sand, Silt, Loess, Loam
Unconsolidated Clay & Organic					Peat		Stratified Clay			Unweathered Clay
Consolidated Rocks	Highly Fractured Rocks				Oil Rocks Reservoir			Sandstone		Limestone		Granite

By combing Eqn. (2)-(4), Darcy’s Law can also be written in terms of the fluid pressure, permeability, and the viscosity

u = ϕ v = - \frac{κ}{μ} (\nabla P - ρ g)

(5)

Here, $\nabla P$ is the fluid pressure gradient. If the gravity term is negligible compared to the pressure gradient, Eqn. (5) can be simplified to

u = - \frac{κ}{μ} \nabla P

(6)

The characteristic time of the advection is the residence time, i.e., how long does a fluid particle stays within a given system. The residence time, $τ_{a}$ , can be calculated as follows:

τ_{a} = \frac{V_{pore}}{Q} = \frac{ϕ L A}{u A} = \frac{ϕ L}{u} = ϕ \frac{L}{u}

(7)

Here V_pore is the pore volume (m³), Q is the flow rate (m³/s), L is the length, A is the cross-section of the porous media in the direction perpendicular to the flow.

Darcy’s Law is applicable at the continuum scale where a representative elementary volume (REV) is significantly larger than the average grain size. The range for the validity of the Darcy’s Law can be checked using the Reynolds number Re:

Re = \frac{4 ρ Q}{p μ}

(8)

where the volumetric flow rate Q (m³/s) and p is the perimeter of a channel or grain size (m). The upper limit of the validity of the Darcy’s Law is when Re is between 1 and 10 [Bear, 2013].