Groundwater flow equation

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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 constituative equation called Darcy's law.

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[edit] Mass Balance

A mass balance must be performed, 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.

\frac{\Delta M_{stor}}{\Delta t} = \frac{M_{in}}{\Delta t} - \frac{M_{out}}{\Delta t} - \frac{M_{gen}}{\Delta t}

[edit] 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 \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 \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 \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

\frac{\partial h}{\partial t} = \alpha\nabla^2 h - G.

[edit] Rectangular Cartesian Coordinates

Three-dimensional finite difference grid used in MODFLOW
Three-dimensional finite difference grid used in MODFLOW

Especially when using rectangular grid finite-difference models (e.g. MODFLOW, made by the USGS), we deal with Cartesian coordinates. In these coordinates the general Laplacian operator becomes (for three-dimensional flow) specifically

\frac{\partial h}{\partial t} = \alpha \left[ \frac{\partial^2 h}{\partial x^2} +\frac{\partial^2 h}{\partial y^2} +\frac{\partial^2 h}{\partial z^2}\right] - G.

As an aside, MODFLOW is actually a "quasi 3D" simulation; it only deals with the vertically averaged T and S, rather than k and Ss. In the PDE solved by MODFLOW there is no vertical (z-direction) derivative, flow is calculated between 2D horizontal layers using the concept of leakage.

[edit] Circular Cylindrical Coordinates

Another useful coordinate system is 3D cylindrical coordinates (typically where a pumping well is a line source located at the origin — parallel to the z axis — causing converging radial flow). Under these conditions the above equation becomes (r being radial distance and θ being angle),

\frac{\partial h}{\partial t} = \alpha \left[ \frac{\partial^2 h}{\partial r^2} + \frac{1}{r} \frac{\partial h}{\partial r} + \frac{1}{r^2} \frac{\partial^2 h}{\partial \theta^2} +\frac{\partial^2 h}{\partial z^2} \right] - G.

[edit] Polar Coordinates

For most typical radial flow problems, there is a symmetry in θ and if the pumping well is fully penetrating (the well is completed across the entire thickness of aquifer), then all flow is horizontal and radial, so the problem becomes 1D (assuming that the pumping well is at the origin). When switching from 3D to 2D (or here 1D) flow, the definition of α changes conceptually (but not in value), now becoming α = T/S (where T is thickness times k and S is thickness times Ss). This simplifies the problem to

\frac{\partial h}{\partial t} = \alpha \left[ \frac{\partial^2 h}{\partial r^2} + \frac{1}{r} \frac{\partial h}{\partial r} \right] - G.

Most of the analytic aquifer test solutions are derived in radial coordinates; a common method for solving the PDE utilizes the Hankel transform. This transform removes the radial dependence of the PDE, leaving an ordinary differential equation in time, t. After solving the transformed algebra problem in Hankel space, the solution is obtained through either an analytic or numerical inverse Hankel transform.

[edit] Assumptions

This equation represents flow to a pumping well (a sink of strength G), located at the origin. Both this equation and the Cartesian version above are the fundamental equation in groundwater flow, but to arrive at this point requires considerable simplification. Some of the main assumptions which went into both these equations are:

  • the aquifer material is incompressible (no change in matrix due to changes in pressure — aka subsidence),
  • the water is of constant density (incompressible),
  • any external loads on the aquifer (e.g., overburden, atmospheric pressure) are constant,
  • for the 1D radial problem the pumping well is fully penetrating a non-leaky aquifer,
  • the groundwater is flowing slowly (Reynolds number less than unity), and
  • the hydraulic conductivity (k) is an isotropic scalar.

Despite these large assumptions, the groundwater flow equation does a good job of representing the distribution of heads in aquifers due to a transient distribution of sources and sinks.

[edit] Laplace Equation (Steady-State Flow)

If the aquifer has recharging boundary conditions a steady-state may be reached (or it may be used as an approximation in many cases), and the diffusion equation (above) simplifies to the Laplace equation.

0 = \alpha\nabla^2 h

This equation states that hydraulic head is a harmonic function, and has many analogs in other fields. The Laplace equation can be solved using techniques, using similar assumptions stated above, but with the additional requirements of a steady-state flow field.

A common method for solution of this equations in civil engineering and soil mechanics is to use the graphical technique of drawing flownets; where contours of hydraulic head and the stream function make a curvilinear grid, allowing complex geometries to be solved approximately.

Steady-state flow to a pumping well (which never truly occurs, but is sometimes a useful approximation) is commonly called the Thiem solution.

[edit] See also

  • Dupuit assumption – a simplification of the groundwater flow equation regarding vertical flow

[edit] Further reading

  • Wang, H.F. and Anderson, M.P. (1982). Introduction to Groundwater Modeling: Finite Difference and Finite Element Methods, 237. ISBN 0-7167-1303-9. 
An excellent beginner's read for groundwater modeling. Covers all the basic concepts, with simple examples in FORTRAN 77.

[edit] External links

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