Physical processes and governing equations

Highlights
Description of major physical processes in limestone aquifers Governing equations for flow and transport in limestone

Physical processes Flow in fractured limestone can be very complex, because strongly contrasting flow conditions between fractures and matrix can prevail. The flow in the fractures can be very fast, whereas it is usually very slow in the limestone matrix. The hydraulic conductivity in the matrix can be orders of magnitude lower than the fracture hydraulic conductivity. The combined hydraulic conductivity of fractures and matrix is called bulk hydraulic conductivity. Transport of substances in porous media can be subdivided in advective transport (due to the groundwater flow) and diffusive/dispersive transport (related to concentration gradients). The fate of contaminants in fractured limestone aquifer is determined by both, transport in fractures and the matrix. The fractures are the primary transport pathways for contaminants, where mainly advective transport dominates. The hydraulic conductivity of the fractures is mainly depending on their aperture (width). For an estimation of the flow velocity in fractures, the cubic law is often used as approximation. With that, the water flux scales with the third power of the fracture aperture. Hence, fractures with a big aperture are far more important as small fractures, since the total water flow and the flow velocity velocity are greater in them. The connectivity of the fractures is another very important parameter for the spreading of a substance in the aquifer. While being transported through the fractures, the contaminant diffuses into the surrounding matrix, which provides usually a high porosity and can store substantial amounts of contaminants. As a consequence, the concentration in the fracture decreases. The continuous diffusion from the fractures into the matrix slows the propagation speed of the plume more and more down. Once diffused into the matrix, it is difficult to remove the contaminant again. Due to the extremely low flow velocities in the limestone matrix, the removal of contaminants from the matrix with a pump-and-treat remediation happens mainly due to back-diffusion from the matrix into the fractures, which can take very long. Part of the contaminant can be sorbed to the surface of the limestone matrix (and the fracture walls), which has a retarding effect on plume propagation. The sorption behavior is often quantified by sorption coefficients or retardation coefficients. Furthermore, contaminants can be degraded by microbial activity, if the proper microbes are present and the conditions are favorable (for PCE: anaerobic conditions). Governing equations The basic equations for flow and transport in fractured porous media are shown below. For more details, be referred to standard literature like e.g. Bear, J. (1972) or Fetter, C.W. (2008) . Flow Groundwater flow in porous media is usually described by Darcy's law, giving a relation between hydraulic head gradient and groundwater flow, with the hydraulic conductivity as proportionality factor. The flow field is calculated by solving the mass balance for incompressible fluid flow in combination with Darcy's law. Mass balance equation for incompressible fluid flow: [math] S_\text{s} \frac{\partial h}{\partial t} - \nabla \cdot (\mathbf{K} \nabla h) = q_\text{w} [/math] Darcy's law (also used in the fractures with the fracture hydraulic conductivity $k_f$) relates water fluxes to hydraulic head gradients: [math] q_\text{darcy} = -\mathbf{K} \nabla h [/math] The hydraulic conductivities in the fractures can be several orders of magnitude higher than the conductivity of the limestone matrix. The fracture conductivity is usually calculated via the hydraulic fracture aperture using the cubic law, which gives a relation between fracture aperture and fracture conductivity. The cubic law considers the fracture flow as flow between parallel plates with no roughness. With the Cubic law, the fracture conductivity can be estimated as: [math] K_\text{f} = \frac{(2b)^2 \rho_\text{w} g}{12 \mu_\text{w}} [/math] Variable Name Unit [math] S_\text{s} [/math] specific storage [math] \text{1/m} [/math] [math] t [/math] time [math] \text{s} [/math] [math] \mathbf{K} [/math] hydraulic conductivity $\text{m/s}$ [math] q_\text{w} [/math] water sources/sinks [math] \text{m}^3\text{/m}^3 [/math] [math] q_\text{darcy} [/math] Darcy flux [math] \text{m}^3/\text{(m}^2 \text{s)} [/math] [math] K_\text{f} [/math] fracture conductivity [math] \text{m/s} [/math] [math] 2b [/math] hydraulic fracture aperture [math] \text{m} [/math] [math] \rho_\text{w} [/math] density of water [math] \text{kg/m}^3 [/math] [math] \mu_\text{w} [/math] dynamic viscosity of water [math] \text{Pa} \cdot \text{s} [/math] [math] g [/math] gravity constant [math] \text{m/s}^2 [/math] Contaminant transport The advection-dispersion equation (ADE) can be used to describe the transport of a solute in a porous medium. Transport equation for a dissolved species: [math](n + \rho_\text{b} k_\text{d}) \frac{\partial c}{\partial t} + \nabla \cdot (n \mathbf{v} c) - \nabla \cdot (n \mathbf{D}_\text{m} \nabla c) = q_\text{m}[/math] The first term describes the storage (here including retardation due to linear sorption), the second and third terms describe advective and dispersive transport of a solute. Degradation can be accounted for as additional source/sink term. Transport in fractures: [math] 2b R_\text{f} \frac{\partial c_\text{f}}{\partial t} + \nabla \cdot (2b \mathbf{v}_\text{f} c_\text{f}) - \nabla \cdot (2b D_\text{f} \nabla_\text{T} c_\text{f}) = q_\text{f}[/math] Variables are: Variable Name Unit [math] n [/math] porosity $ - $ [math] q_\text{m}\text{,} q_\text{f} [/math] mass sources and sinks, also degradation $ \text{kg/(m}^3 \text{s)} $ [math] c_\text{f} [/math] concentration in the fractures $ \text{kg/m}^3 $ [math] \rho_\text{b} [/math] bulk density $ \text{kg/m}^3 $ [math] k_\text{d} [/math] sorption coefficient $ \text{m}^3\text{/kg} $ [math] c [/math] concentration $ \text{kg/m}^3 $ [math] \mathbf{v} [/math] flow velocity in the matrix $ \text{m/s} $ [math] \mathbf{v}_\text{f} [/math] flow velocity in the fractures $ \text{m/s} $ [math] R_\text{f} [/math] retardation factor accounting for sorption to the fracture walls $ 1 $ [math] \mathbf{D}_\text{m}\text{,} D_\text{f} [/math] hydrodynamic dispersion tensor and coefficient $ \text{m}^2\text{/s} $ [math] \nabla_\text{T} [/math] tangential gradient $ \text{1/m} $ Coupling of flow and transport between fractures and matrix The fracture and the matrix system are coupled by the continuity of water fluxes and solute fluxes at the fracture-matrix interface. Furthermore, the hydraulic head and the solute concentration are continuous at the interface between fractures and matrix. Return to Content