Groundwater Velocity Partition

By Darrel Dunn, Ph,D., PG, Hydrogeologist-Geologist

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The natural groundwater velocity at any point in the subsurface may be thought of as the sum of all the partial velocities resulting from all the individual features belonging to a hierarchy of features of the configuration of the water table.  These velocities are dependent upon the interaction of the water table configuration and the hydraulic conductivity variations of the system.  The elements of the hierarchy of features may be designated in any convenient way.  Perhaps the simplest way is to assign names to them.  This is easily done in areas where the water table follows the land surface and the topographic features of the land surface have names.  In this case the topographic names may also be used for the associated water table features.  Another way to designate the features is to classify them by grouping features of similar size and shape.  For example water table mounds associated with major mountain chains might be designated as first order features, those associated with major ranges of high hills could be called second order features, medium sized hills might correspond to a third order, and successively smaller topographic features might give rise to fourth and fifth orders

Recall that Darcy's Law at a location in a groundwater system is V = -K grad h, where V is the specific dischage vector (hereafter called the groundwater velocity vector), K is the tensor of hydraulic conductivity, and grad h is the hydraulic head vector.  Let Vn be a natural groundwater velocity vector in a system in which five orders of water table mounds have been defined.   Vn is a function of position in space and time.  Let V1234 be the velocity which would occur in the same five order system with the nth water table features removed.  Then a fifth order partial velocity may be defined as V5 = Vn - V1234.  This velocity is the velocity due to the presence of fifth order water table reatures.  Similarly,

V4 = V1234 - V123.  

V3 = V123 - V12

 V2 = V12 - V1

where V123 is the velocity of the system with the fourth and fifth order features removed, and V12 and V1 are similarly defined.

Observe that

 V1 + V2 +  V3 + V4 + V5 = Vn .

For each order of water table feature there is a velocity which results from its presence superimposed upon the higher order features, and the sum of these velocities is the natural velocity of the water at a point in the system.

The true velocity V of the groundwater at a point is not necessarily the natural velocity Vn because of the effects of human interference.  In this case an altered velocity may be defined as Va = V -Vn.