4.3 Transport Processes in Groundwater

Vincent E.A. Post; Craig T. Simmons

4.3 Transport Processes in Groundwater

Before continuing this discussion, it is useful to review some of the most important terms related to groundwater transport processes, as there can be confusion in some cases. Hydrogeologists generally refer to the movement of groundwater simply as flow. In fluid mechanics, from which many insights about variable-density phenomena are derived, the flow of a fluid is referred to as advection. In hydrogeology, this term is generally used in a stricter sense and is understood to refer to the transport of solutes by flowing groundwater. Groundwater can also transport heat, and this is called convection, although the term is used quite liberally and can refer to advective flow as well. In the context of variable-density flow, the term convection tends to be used to indicate flow that arises as a consequence of density variations.

As solutes or heat are transported, initially steep concentration or temperature gradients become more diffuse. Spreading of solutes due to concentration differences is called diffusion. The equivalent process for heat transport driven by temperature gradients is called conduction.

The solute mass flux per unit area of the bulk porous medium due to diffusion is given by Fick’s law and presented in Equation 21.

$\displaystyle \vec{J}_{C}=-D_{C}\nabla C$

(21)

where:

$\vec{J}_{C}$	=	solute mass flux (M/(L²T)), e.g., kg m⁻² s⁻¹
D_C	=	solute diffusion coefficient in the porous medium (L²/T), e.g., m² s⁻¹
C	=	solute concentration (M/L³), e.g., kg m⁻³

The porous medium diffusion coefficient D_C is lower than the diffusion coefficient in free water because the solutes have to move along a tortuous flow path in the connected pore space. For conduction, the heat (or energy) flux per unit area of the bulk porous medium is given by Fourier’s law (Equation 22).

$\displaystyle \vec{J}_{e}=-c_{P}D_{T}\nabla T=-k_{T}\nabla T$

(22)

where:

$\vec{J}_{e}$	=	heat flux (M/T³), e.g., kg s⁻³, or W m⁻²
c_P	=	volumetric heat capacity (M/(LT²Θ)), e.g., kg² m⁻¹s⁻² K⁻¹
D_T	=	thermal diffusivity (L²/T), e.g., m² s⁻¹
T	=	temperature (Θ), e.g., K
k_T	=	thermal conductivity (M/(LΘ)), e.g., W m⁻¹ K⁻¹, k_T = c_PD_T

Just like Equation 5, the flux in Equations 21 and 22 is the product of a proportionality constant and the gradient of a field variable. Because of their resemblance, these relationships are all called Fickian type equations or diffusion equations. There is a fundamental difference between solute and heat transfer because solutes can only move through the connected water-filled pore space, but heat is transmitted by conduction through the bulk, wet rock. That means that ${\vec{J}}_C$ must be multiplied by the porosity n to obtain the net solute mass flux, i.e., the transported mass of solute per unit of surface area per unit of time. This is not necessary for the heat flux ${\vec{J}}_e$ as long as k_T represents the bulk thermal conductivity (i.e., the net value for the groundwater and the rock material).

As Equations 21 and 22 show, diffusion and conduction operate when there is a concentration ( $\nabla C$ ) or temperature ( $\nabla T$ ) gradient, and assume no forced advection (i.e., there need not be physical movement of groundwater through the porous medium). Diffusion and conduction can be regarded as a “spreading” process in a system that is hydraulically stagnant. When groundwater is flowing, solutes will migrate in the flowing groundwater across a range of spatial scales while at the same time spreading by diffusion (or in the case of heat, conduction). At the pore scale, water flows fastest through the pore centers but is stagnant at the pore wall. At the scale of a pore network, the flow path of water is tortuous due to the irregular shape of the connected pore space. At the scale of an aquifer, differences in permeability cause water to flow fast in some zones, and slow in others. These velocity variations at various scales cause spreading, which is referred to as mechanical dispersion. For most groundwater systems, the effects of mechanical dispersion tend to dominate over diffusion.

In combination, the combined process of diffusion and mechanical dispersion is called hydrodynamic dispersion, D_H. Hydrodynamic dispersion is commonly described in transport models using Equation 21, albeit with a hydrodynamic dispersion tensor replacing the diffusion coefficient D_C. The coefficients of mechanical dispersion are functionally related to the groundwater seepage velocity (that is, they have higher values for higher flow velocities) (Bear, 1972).

Flow driven by hydraulic head gradients is commonly called forced convection. Truly free convection, as in the Simmons and others (2002b) laboratory experiment of Figure 8 or the unstable heat flow example in Figure 9b, is rare in groundwater systems, where density-driven flow mostly occurs in the presence of background groundwater flow. A flow regime influenced by both forced and free convection is called mixed convection. Understanding the magnitude of the different driving forces in any given hydrogeological situation is critical, as is the need to appreciate the different flow processes that can occur in a variable-density flow system. These will affect the magnitude of the solute fluxes and the patterns and rates of plume migration. Comparison of time-length scales for the flow and transport processes discussed above can be useful when assessing the importance of density-driven flow in groundwater (Section 5).

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