Used in hydrogeology, the groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through an aquifer. The transient flow of groundwater is described by a form of the diffusion equation, similar to that used in heat transfer to describe the flow of heat in a solid (heat conduction). The steady-state flow of groundwater is described by a form of the Laplace equation, which is a form of potential flow and has analogs in numerous fields. The groundwater flow equation is often derived for a small representative elemental volume (REV), where the properties of the medium are assumed to be effectively constant. A mass balance is done on the water flowing in and out of this small volume, the flux terms in the relationship being expressed in terms of head by using the constitutive equation called Darcy's law, which requires that the flow is laminar. Other approaches are based on Agent Based Models to incorporate the effect of complex aquifers such as karstic or fractured rocks (i.e. volcanic)
Mass balance A mass balance must be performed, and used along with Darcy's law, to arrive at the transient groundwater flow equation. This balance is analogous to the energy balance used in heat transfer to arrive at the heat equation. It is simply a statement of accounting, that for a given control volume, aside from sources or sinks, mass cannot be created or destroyed. The conservation of mass states that, for a given increment of time (Δt), the difference between the mass flowing in across the boundaries, the mass flowing out across the boundaries, and the sources within the volume, is the change in storage.
Δ M s t o r Δ t = M i n Δ t − M o u t Δ t − M g e n Δ t {\displaystyle {\frac {\Delta M_{stor}}{\Delta t}}={\frac {M_{in}}{\Delta t}}-{\frac {M_{out}}{\Delta t}}-{\frac {M_{gen}}{\Delta t}}}
Diffusion equation (transient flow) Mass can be represented as density times volume, and under most conditions, water can be considered incompressible (density does not depend on pressure). The mass fluxes across the boundaries then become volume fluxes (as are found in Darcy's law). Using Taylor series to represent the in and out flux terms across the boundaries of the control volume, and using the divergence theorem to turn the flux across the boundary into a flux over the entire volume, the final form of the groundwater flow equation (in differential form) is:
S s ∂ h ∂ t = − ∇ ⋅ q − G . {\displaystyle S_{s}{\frac {\partial h}{\partial t}}=-\nabla \cdot q-G.}
This is known in other fields as the diffusion equation or heat equation, it is a parabolic partial differential equation (PDE). This mathematical statement indicates that the change in hydraulic head with time (left hand side) equals the negative divergence of the flux (q) and the source terms (G). This equation has both head and flux as unknowns, but Darcy's law relates flux to hydraulic heads, so substituting it in for the flux (q) leads to
S s ∂ h ∂ t = − ∇ ⋅ ( − K ∇ h ) − G . {\displaystyle S_{s}{\frac {\partial h}{\partial t}}=-\nabla \cdot (-K\nabla h)-G.}
Now if hydraulic conductivity (K) is spatially uniform and isotropic (rather than a tensor), it can be taken out of the spatial derivative, simplifying them to the Laplacian, this makes the equation
S s ∂ h ∂ t = K ∇ 2 h − G . {\displaystyle S_{s}{\frac {\partial h}{\partial t}}=K\nabla ^{2}h-G.}
Dividing through by the specific storage (Ss), puts hydraulic diffusivity (α = K/Ss or equivalently, α = T/S) on the right hand side. The hydraulic diffusivity is proportional to the speed at which a finite pressure pulse will propagate through the system (large values of α lead to fast propagation of signals). The groundwater flow equation then becomes
∂ h ∂ t = α ∇ 2 h − G . {\displaystyle {\frac {\partial h}{\partial t}}=\alpha \nabla ^{2}h-G.}
Where the sink/source term, G, now has the same units but is divided by the appropriate storage term (as defined by the hydraulic diffusivity substitution).
Rectangular cartesian coordinates
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